(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Fluid Mechanics"

Landau 
Lifshitz 



Fluid Mechanics 

Third Revised English Edition 



c 

IB • 

Z 



3 

n 



rfi I 



Course of Theoretical Physics 
Volume 6 



L. D. Landau (Deceased) and E. M. Lifshitz 

Institute of Physical Problems 
USSR Academy of Sciences 



gamon 





CVII 



COURSE OF THEORETICAL PHYSICS 
Volume 6 

FLUID MECHANICS 



COURSE OF THEORETICAL PHYSICS 

Vol. 1 Mechanics 

Vol. 2 The Classical Theory of Fields 

Vol. 3 Quantum Mechanics— Non-Relativistic Theory 

Vol. 4 Relativistic Quantum Theory 

Vol. 5 Statistical Physics 

Vol. 7 Tfceory o/ Elasticity 

Vol. 8 Electrodynamics of Continuous Media 

Vol. 9 Physical Kinetics 



FLUID MECHANICS 



by 
L. D. LANDAU and E. M. LIFSHITZ 

INSTITUTE OF PHYSICAL PROBLEMS, U.S.S.R. ACADEMY OF SCIENCES 

Volume 6 of Course of Theoretical Physics 

Translated from the Russian by 
J. B. SYKES and W. H. REID 



PERGAMON PRESS 

Oxford • London • Edinburgh • New York 
Toronto • Sydney • Paris • Braunschweig 



Pergamon Press Ltd. , Headington Hill Hall, Oxford 

4 & 5 Fitzroy Square, London W.l 

Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 

Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101 

Pergamon of Canada, Ltd., 6 Adelaide Street East, Toronto, Ontario 

Pergamon Press (Aust.) Pty. Ltd., 20-22 Margaret Street, Sydney, N.S.W. 

Pergamon Press S.A.R.L., 24 rue des Ecoles, Paris 5 e 

Vieweg & Sohn GmbH., Burgplatz 1, Braunschweig 



Copyright 

© 
1959 

Pergamon Press Ltd. 

First published in English 1959 
Second impression 1963 
Third impression 1966 



Sole distributors in the U.S.A., Addison- Wesley Publishing Company, Inc. 
Reading, Massachusetts • Palo Alto ■ London • Don Mills, Ontario 



Library of Congress Card Number 59-10525 



Printed in Great Britain by J. W. Arrovismiih Ltd., Bristol 

330/59 



CONTENTS 

Page 

Preface to the English edition xi 

Notation x ii 

I. IDEAL FLUIDS 

§1. The equation of continuity 1 

§2. Euler's equation 2 

§3. Hydrostatics 6 

§4. The condition that convection is absent 8 

§5. Bernoulli's equation 9 

§6. The energy flux 10 

§7. The momentum flux 12 

§8 The conservation of circulation 14 

§9. Potential flow 16 

§10. Incompressible fluids 20 

§11. The drag force in potential flow past a body 31 

§12. Gravity waves 36 

§13. Long gravity waves 42 

§14. Waves in an incompressible fluid 44 

II. VISCOUS FLUIDS 

§15. The equations of motion of a viscous fluid 47 

§16. Energy dissipation in an incompressible fluid 53 

§17. Flow in a pipe 55 

§18. Flow between rotating cylinders 60 

§19. The law of similarity 61 

§20. Stokes' formula 63 

§21. The laminar wake 71 

§22. The viscosity of suspensions 76 
§23. Exact solutions of the equations of motion for a viscous fluid 79 

§24. Oscillatory motion in a viscous fluid 88 

§25. Damping of gravity waves 98 

III. TURBULENCE 

§26. Stability of steady flow 102 

§27. The onset of turbulence 103 

§28. Stability of flow between rotating cylinders 107 

§29. Stability of flow in a pipe 1 1 1 



vi Contents 

Page 

§30. Instability of tangential discontinuities 114 

§31. Fully developed turbulence 116 

§32. Local turbulence 120 

§33. The velocity correlation 123 

§34. The turbulent region and the phenomenon of separation 128 

§35. The turbulent jet 130 

§36. The turbulent wake 136 

§37. Zhukovskii's theorem 137 

§38. Isotropic turbulence 140 

IV. BOUNDARY LAYERS 

§39. The laminar boundary layer 145 

§40. Flow near the line of separation 151 

§41. Stability of flow in the laminar boundary layer 156 

§42. The logarithmic velocity profile 159 

§43. Turbulent flow in pipes 163 

§44. The turbulent boundary layer 166 

§45. The drag crisis 168 

§46. Flow past streamlined bodies 172 

§47. Induced drag 175 

§48. The lift of a thin wing 179 

V. THERMAL CONDUCTION IN FLUIDS 

§49. The general equation of heat transfer 183 

§50. Thermal conduction in an incompressible fluid 188 

§51. Thermal conduction in an infinite medium 192 

§52. Thermal conduction in a finite medium 196 

§53. The similarity law for heat transfer 202 

§54. Heat transfer in a boundary layer 205 

§55. Heating of a body in a moving fluid 209 

§56. Free convection 212 

VI. DIFFUSION 

§57. The equations of fluid dynamics for a mixture of fluids 219 

§58. Coefficients of mass transfer and thermal diffusion 222 

§59. Diffusion of particles suspended in a fluid 227 

VII. SURFACE PHENOMENA 

§60. Laplace's formula 230 

§61. Capillary waves 237 

§62. The effect of adsorbed films on the motion of a liquid 241 



Contents vii 

VIII. SOUND Page 

§63. Sound waves 245 

§64. The energy and momentum of sound waves 249 

§65. Reflection and refraction of sound waves 253 

§66. Geometrical acoustics 256 

§67. Propagation of sound in a moving medium 259 

§68. Characteristic vibrations 262 

§69. Spherical waves 265 

§70. Cylindrical waves 268 

§71. The general solution of the wave equation 270 

§72. The lateral wave 273 

§73. The emission of sound 279 

§74. The reciprocity principle 288 

§75. Propagation of sound in a tube 291 

§76. Scattering of sound 294 

§77. Absorption of sound 298 

§78. Second viscosity 304 

IX. SHOCK WAVES 

§79. Propagation of disturbances in a moving gas 310 

§80. Steady flow of a gas 313 

§81. Surfaces of discontinuity 317 

§82. The shock adiabatic 319 

§83. Weak shock waves 322 

§84. The direction of variation of quantities in a shock wave 325 

§85. Shock waves in a perfect gas 329 

§86. Oblique shock waves 333 

§87. The thickness of shock waves 337 

§88. The isothermal discontinuity 342 

§89. Weak discontinuities 344 

X. ONE-DIMENSIONAL GAS FLOW 

§90. Flow of gas through a nozzle 347 

§91. Flow of a viscous gas in a pipe 350 

§92. One-dimensional similarity flow 353 

§93. Discontinuities in the initial conditions 360 

§94. One-dimensional travelling waves 366 

§95. Formation of discontinuities in a sound wave 372 

§96. Characteristics 373 

§97. Riemann invariants 381 

§98. Arbitrary one-dimensional gas flow 386 

§99. The propagation of strong shock waves 392 

§100. Shallow-water theory 396 



viii Contents 

XL THE INTERSECTION OF SURFACES OF DISCONTINUITY 

Page 

§101. Rarefaction waves 399 

§102. The intersection of shock waves 405 

§103. The intersection of shock waves with a solid surface 410 

§104. Supersonic flow round an angle 413 

§105. Flow past a conical obstacle 418 

XII. TWO-DIMENSIONAL GAS FLOW 

§106. Potential flow of a gas 422 
§107. Steady simple waves 425 
§108. Chaplygin's equation: the general problem of steady two- 
dimensional gas flow 430 
§109. Characteristics in steady two-dimensional flow 433 
§110. The Euler-Tricomi equation. Transonic flow 436 
§111. Solutions of the Euler-Tricomi equation near non-singular 

points of the sonic surface 441 

§1 12. Flow at the velocity of sound 446 

§113. The intersection of discontinuities with the transition line 451 

XIII. FLOW PAST FINITE BODIES 

§114. The formation of shock waves in supersonic flow past bodies 457 

§115. Supersonic flow past a pointed body 460 

§116. Subsonic flow past a thin wing 464 

§117. Supersonic flow past a wing 466 

§118. The law of transonic similarity 469 

§119. The law of hypersonic similarity 472 

XIV. FLUID DYNAMICS OF COMBUSTION 

§120. Slow combustion 474 

§121. Detonation 480 

§122. The propagation of a detonation wave 487 

§123. The relation between the different modes of combustion 493 

§124. Condensation discontinuities 496 

XV. RELATIVISTIC FLUID DYNAMICS 

§125. The energy-momentum tensor 499 

§126. The equations of relativistic fluid dynamics 500 

§127. Relativistic equations for dissipative processes 505 



Contents ix 

XVI. DYNAMICS OF SUPERFLUIDS Page 

§128. Principal properties of superfluids 507 

§129. The thermo-mechanical effect 509 

§130. The equations of superfluid dynamics 510 

§131. The propagation of sound in a superfluid 517 

XVII. FLUCTUATIONS IN FLUID DYNAMICS 

§132. The general theory of fluctuations in fluid dynamics 523 

§133. Fluctuations in an infinite medium 526 

Index 530 



PREFACE TO THE ENGLISH EDITION 

The present book deals with fluid mechanics, i.e. the theory of the motion of 
liquids and gases. 

The nature of the book is largely determined by the fact that it describes 
fluid mechanics as a branch of theoretical physics, and it is therefore markedly 
different from other textbooks on the same subject. We have tried to develop 
as fully as possible all matters of physical interest, and to do so in such a way 
as to give the clearest possible picture of the phenomena and their interrela- 
tion. Accordingly, we discuss neither approximate methods of calculation in 
fluid mechanics, nor empirical theories devoid of physical significance. On 
the other hand, accounts are given of some topics not usually found in text- 
books on the subject: the theory of heat transfer and diffusion in fluids; 
acoustics; the theory of combustion; the dynamics of superfluids; and 
relativistic fluid dynamics. 

In a field which has been so extensively studied as fluid mechanics it was 
inevitable that important new results should have appeared during the several 
years since the last Russian edition was published. Unfortunately, our 
preoccupation with other matters has prevented us from including these 
results in the English edition. We have merely added one further chapter, 
on the general theory of fluctuations in fluid dynamics. 

We should like to express our sincere thanks to Dr Sykes and Dr Reid 
for their excellent translation of the book, and to Pergamon Press for their 
ready agreement to our wishes in various matters relating to its publication. 

Moscow L. D. Landau 

E. M. Lifshitz 



NOTATION 

p density 
p pressure 
T temperature 
s entropy per unit mass 
e internal energy per unit mass 
w = e +pjp heat function (enthalpy) 
7 = c v jc v ratio of specific heats at constant pressure and constant 

volume 
rj dynamic viscosity 
v = rjjp kinematic viscosity 
k thermal conductivity 
X = KJpCp thermometric conductivity 
R Reynolds number 
c velocity of sound 
M ratio of fluid velocity to velocity of sound 



CHAPTER I 



IDEAL FLUIDS 



§1. The equation of continuity 

Fluid dynamics concerns itself with the study of the motion of fluids 
(liquids and gases). Since the phenomena considered in fluid dynamics are 
macroscopic, a fluid is regarded as a continuous medium. This means that 
any small volume element in the fluid is always supposed so large that it still 
contains a very great number of molecules. Accordingly, when we speak 
of infinitely small elements of volume, we shall always mean those which are 
"physically" infinitely small, i.e. very small compared with the volume of 
the body under consideration, but large compared with the distances between 
the molecules. The expressions fluid particle and point in a fluid are to be 
understood in a similar sense. If, for example, we speak of the displacement 
of some fluid particle, we mean not the displacement of an individual mole- 
cule, but that of a volume element containing many molecules, though still 
regarded as a point. 

The mathematical description of the state of a moving fluid is effected by 
means of functions which give the distribution of the fluid velocity 
v = v(x, y, z, t) and of any two thermodynamic quantities pertaining to 
the fluid, for instance the pressure p(x, y, z, t) and the density p(x, y, z, t). 
As is well known, all the thermodynamic quantities are determined by the 
values of any two of them, together with the equation of state; hence, if 
we are given five quantities, namely the three components of the velocity v, 
the pressure p and the density p, the state of the moving fluid is completely 
determined. 

All these quantities are, in general, functions of the co-ordinates x, y, z 
and of the time t. We emphasise that v(x, y, z y t) is the velocity of the 
fluid at a given point (x, y, z) in space and at a given time t , i.e. it refers to 
fixed points in space and not to fixed particles of the fluid ; in the course of 
time, the latter move about in space. The same remarks apply to p and p. 

We shall now derive the fundamental equations of fluid dynamics. Let 
us begin with the equation which expresses the conservation of matter. 
We consider some volume Vq of space. The mass of fluid in this volume 
is / p d V, where p is the fluid density, and the integration is taken over the 
volume Vo. The mass of fluid flowing in unit time through an element df 
of the surface bounding this volume is pv • df ; the magnitude of the vector 
df is equal to the area of the surface element, and its direction is along the 
normal. By convention, we take df along the outward normal. Then p\ • df 
is positive if the fluid is flowing out of the volume, and negative if the flow 

1 



2 Ideal Fluids §2 

is into the volume. The total mass of fluid flowing out of the volume Vq 
in unit time is therefore 



<j> pv»df, 



where the integration is taken over the whole of the closed surface surround- 
ing the volume in question. 

Next, the decrease per unit time in the mass of fluid in the volume Vq 
can be written 



8t 
Equating the two expressions, we have 

d 
~dt 



! 



8 . 

pdV. 



fpdF = - (jjpv.df. (1.1) 



The surface integral can be transformed by Green's formula to a volume 
integral: 



(J) p v* df = div (pv) dV. 



Thus 

dp 



J[^ + div(pv)]dF=0. 



Since this equation must hold for any volume, the integrand must vanish, 
i.e. 



This is the equation of continuity. Expanding the expression div (pv), we 
can also write (1.2) as 



dp/dt + div (pv) = 0. (1.2) 

ntinuity. Expanding the expression div (pv), we 

dpjdt+p div v+vgradp = 0. (1.3) 

The vector 

J = pv (1.4) 

is called the mass flux density. Its direction is that of the motion of the 
fluid, while its magnitude equals the mass of fluid flowing in unit time 
through unit area perpendicular to the velocity. 

§2. Euler's equation 

Let us consider some volume in the fluid. The total force acting on this 
volume is equal to the integral 

- jpdf 



§2 Euler's equation 3 

of the pressure, taken over the surface bounding the volume. Transforming 
it to a volume integral, we have 

- <j)/>df = - j gradpdV. 

Hence we see that the fluid surrounding any volume element dV exerts 
on that element a force -dVgradp. In other words, we can say that a 
force — gradp acts on unit volume of the fluid. 

We can now write down the equation of motion of a volume element in 
the fluid by equating the force -gmdp to the product of the mass per unit 
volume (/>) and the acceleration dvjdt: 

p dvjdt = -grad/>. (2.1) 

The derivative dv/dt which appears here denotes not the rate of change 
of the fluid velocity at a fixed point in space, but the rate of change of the 
velocity of a given fluid particle as it moves about in space. This derivative 
has to be expressed in terms of quantities referring to points fixed in space. 
To do so, we notice that the change dv in the velocity of the given fluid 
particle during the time dt is composed of two parts, namely the change 
during d* in the velocity at a point fixed in space, and the difference between 
the velocities (at the same instant) at two points dr apart, where dr is the 
distance moved by the given fluid particle during the time dt. The first 
part is (dv/dt)dt f where the derivative dv/dt is taken for constant x, y, z, 
i.e. at the given point in space. The second part is 

dv dv dv t% 

dx— + dv— + dz— = (dr-grad)v. 

dx dy dz 

Thus 

dv = (dv/d*)d* + (dr-grad)v, 

or, dividing both sides by dt, 

— = — +(vgrad)v. (2.2) 

dt dt K 

Substituting this in (2.1), we find 

dv 1 

— + (v«grad)v = gradp. (2.3) 

dt p 

This is the required equation of motion of the fluid; it was first obtained 
by L. Euler in 1755. It is called Euler's equation and is one of the funda- 
mental equations of fluid dynamics. 

If the fluid is in a gravitational field, an additional force />g, where g 
is the acceleration due to gravity, acts on any unit volume. This force 



4 Ideal Fluids §2 

must be added to the right-hand side of equation (2.1), so that equation (2.3) 
takes the form 

— + (v.grad)v = - 5 L + g . (2 . 4 ) 

ot p 

In deriving the equations of motion we have taken no account of processes 
of energy dissipation, which may occur in a moving fluid in consequence of 
internal friction (viscosity) in the fluid and heat exchange between different 
parts of it. The whole of the discussion in this and subsequent sections of 
this chapter therefore holds good only for motions of fluids in which thermal 
conductivity and viscosity are unimportant; such fluids are said to be ideal. 

The absence of heat exchange between different parts of the fluid (and 
also, of course, between the fluid and bodies adjoining it) means that the 
motion is adiabatic throughout the fluid. Thus the motion of an ideal 
fluid must necessarily be supposed adiabatic. 

In adiabatic motion the entropy of any particle of fluid remains constant 
as that particle moves about in space. Denoting by s the entropy per unit 
mass, we can express the condition for adiabatic motion as 

dsjdt = 0, (2.5) 

where the total derivative with respect to time denotes, as in (2.1), the rate 
of change of entropy for a given fluid particle as it moves about. This 
condition can also be written 

ds/dt+v-grads *= 0. (2.6) 

This is the general equation describing adiabatic motion of an ideal fluid. 
Using (1.2), we can write it as an "equation of continuity" for entropy: 

d(ps)jdt+div(psv) = 0. (2.7) 

The product psv is the "entropy flux density". 

It must be borne in mind that the adiabatic equation usually takes a much 
simpler form. If, as usually happens, the entropy is constant throughout 
the volume of the fluid at some initial instant, it retains everywhere the same 
constant value at all times and for any subsequent motion of the fluid. 
In this case we can write the adiabatic equation simply as 

s = constant, (2.8) 

and we shall usually do so in what follows. Such a motion is said to be 
isentropic. 

We may use the fact that the motion is isentropic to put the equation of 
motion (2.3) in a somewhat different form. To do so, we employ the 
familiar thermodynamic relation 

dw = Tds+ Vdp, 

where w is the heat function per unit mass of fluid (enthalpy), V = \jp 



§2 Euler's equation 5 

is the specific volume, and T is the temperature. Since s = constant, we 
have simply 

dzv = Vdp = dp/p, 

and so (grad p)jp = grad w. Equation (2.3) can therefore be written in 
the form 

dv/d* + (vgrad)v = -gradw. (2.9) 

It is useful to notice one further form of Euler's equation, in which it in- 
volves only the velocity. Using a formula well known in vector analysis, 

| grad v* = vxcurlv+(v»grad)v, 

we can write (2.9) in the form 

8v/dt+%gradv 2 -YXC\irlv = -gradw. (2.10) 

If we take the curl of both sides of this equation, we obtain 

8 

— (curlv) = curl (v x curl v), (2.11) 

8t 

which involves only the velocity. 

The equations of motion have to be supplemented by the boundary con- 
ditions that must be satisfied at the surfaces bounding the fluid. For an 
ideal fluid, the boundary condition is simply that the fluid cannot penetrate 
a solid surface. This means that the component of the fluid velocity normal 
to the bounding surface must vanish if that surface is at rest: 

v» = 0. (2.12) 

In the general case of a moving surface, v n must be equal to the correspond- 
ing component of the velocity of the surface. 

At a boundary between two immiscible fluids, the condition is that the 
pressure and the velocity component normal to the surface of separation 
must be the same for the two fluids, and each of these velocity components 
must be equal to the corresponding component of the velocity of the 
surface. 

As has been said at the beginning of §1, the state of a moving fluid is 
determined by five quantities : the three components of the velocity v and, 
for example, the pressure p and the density p. Accordingly, a complete 
system of equations of fluid dynamics should be five in number. For an 
ideal fluid these are Euler's equations, the equation of continuity, and 
the adiabatic equation. 

PROBLEM 

Write down the equations for one-dimensional motion of an ideal fluid in terms of the 
variables a, t, where a (called a Lagrangian variable^) is the * co-ordinate of a fluid particle 
at some instant t = t . 



t Although such variables are usually called Lagrangian, it should be mentioned that the equations 
of motion in these co-ordinates were first obtained by Euler, at the same time as equations (2.3). 



6 Ideal Fluids §3 

Solution. In these variables the co-ordinate * of any fluid particle at any instant is re- 
garded as a function of t and its co-ordinate a at the initial instant: x = x(a, t). The condition 
of conservation of mass during the motion of a fluid element (the equation of continuity) 
is accordingly written p dx = p da, or 

8x\ 
P' 



\Sa/ L 



where p {a) is a given initial density distribution. The velocity of a fluid particle is, by 
definition, y = (dx[dt) a , and the derivative (8vjdt) a gives the rate of change of the velocity 
of the particle during its motion. Euler's equation becomes 



/8v\ 



and the adiabatic equation is 



dv\ 1 /dp\ 

8t J a po \da)t 

{ds/dt) a = 0. 



§3. Hydrostatics 

For a fluid at rest in a uniform gravitational field, Euler's equation (2.4) 
takes the form 

gradp = pg. (3.1) 

This equation describes the mechanical equilibrium of the fluid. (If there 
is no external force, the equation of equilibrium is simply gradp = 0, 
i.e. p = constant; the pressure is the same at every point in the fluid.) 

Equation (3.1) can be integrated immediately if the density of the fluid 
may be supposed constant throughout its volume, i.e. if there is no signi- 
ficant compression of the fluid under the action of the external force. Taking 
the -sr-axis vertically upward, we have 

dp/dx = dpjdy = 0, Bpjdz = -pg. 
Hence 

P — —pg z + constant. 

If the fluid at rest has a free surface at height h, to which an external pressure 
po, the same at every point, is applied, this surface must be the horizontal 
plane z = h. From the condition^) = p for z = h, we find that the constant 
is po + pgh y so that 

p=po+ P g(h-z). (3.2) 

For large masses of liquid, and for a gas, the density p cannot in general 
be supposed constant; this applies especially to gases (for example, the 
atmosphere). Let us suppose that the fluid is not only in mechanical 
equilibrium but also in thermal equilibrium. Then the temperature is the 



§3 Hydrostatics 7 

same at every point, and equation (3.1) may be integrated as follows. We 
use the familiar thermodynamic relation 

d<D = -sdT+Vdp, 

where <X> is the thermodynamic potential per unit mass. For constant tem- 
perature 

d0> = Vdp = dpi p. 

Hence we see that the expression (grad p)jp can be written in this case as 
grad O, so that the equation of equilibrium (3.1) takes the form 

grad$ = g. 

For a constant vector g directed along the negative sr-axis we have 

g = -grades). 

Thus 

grad(0+#sr) = 0, 

whence we find that throughout the fluid 

<!?+gz = constant; (3.3) 

gz is the potential energy of unit mass of fluid in the gravitational field. 
The condition (3.3) is known from statistical physics to be the condition 
for thermodynamic equilibrium of a system in an external field. 

We may mention here another simple consequence of equation (3.1). 
If a fluid (such as the atmosphere) is in mechanical equilibrium in a gravi- 
tational field, the pressure in it can be a function only of the altitude z 
(since, if the pressure were different at different points with the same alti- 
tude, motion would result). It then follows from (3.1) that the density 

1 dp 
gdz 

is also a function of z only. The pressure and density together determine 
the temperature, which is therefore again a function of z only. Thus, in 
mechanical equilibrium in a gravitational field, the pressure, density and 
temperature distributions depend only on the altitude. If, for example, the 
temperature is different at different points with the same altitude, then 
mechanical equilibrium is impossible. 

Finally, let us derive the equation of equilibrium for a very large mass of 
fluid, whose separate parts are held together by gravitational attraction — 
a star. Let <£ be the Newtonian gravitational potential of the field due to 
the fluid. It satisfies the differential equation 

A<f> = 47rG/>, (3.5) 



8 Ideal Fluids 



§4 



where G is the Newtonian constant of gravitation. The gravitational accelera- 
tion is -grad <f>, and the force on a mass p is ~ P grad <f>. The condition 
of equilibrium is therefore 

gradp = -pgrad<f>. 

Dividing both sides by P , taking the divergence of both sides, and using 
equation (3.5), we obtain 

div|-gradpj = -4ttG p . (3.6) 

It must be emphasised that the present discussion concerns only mechanical 
equilibrium; equation (3.6) does not presuppose the existence of complete 
thermal equilibrium. 

If the body is not rotating, it will be spherical when in equilibrium, 
and the density and pressure distributions will be spherically symmetrical. 
Equation (3.6) in spherical co-ordinates then takes the form 

1 d / r 2 dp \ 

§4. The condition that convection is absent 

A fluid can be in mechanical equilibrium (i.e. exhibit no macroscopic 
motion) without being in thermal equilibrium. Equation (3.1), the condi- 
tion for mechanical equilibrium, can be satisfied even if the temperature is 
not constant throughout the fluid. However, the question then arises of 
the stability of such an equilibrium. It is found that the equilibrium is 
stable only when a certain condition is fulfilled. Otherwise, the equilibrium 
is unstable, and this leads to the appearance in the fluid of currents which 
tend to mix the fluid in such a way as to equalise the temperature. This 
motion is called convection. Thus the condition for a mechanical equilibrium 
to be stable is the condition that convection is absent. It can be derived as 
follows. 

Let us consider a fluid element at height z, having a specific volume 
V(p, s), where p and s are the equilibrium pressure and entropy at height 
z. Suppose that this fluid element undergoes an adiabatic upward displace- 
ment through a small interval g; its specific volume then becomes V(p\ s), 
where p' is the pressure at height z + £. For the equilibrium to be stable, it 
is necessary (though not in general sufficient) that the resulting force on 
the element should tend to return it to its original position. This means 
that the element must be heavier than the fluid which it "displaces" in its 
new position. The specific volume of the latter is V(p\ s'), where s' is the 
equilibrium entropy at height z+$. Thus we have the stability condition 

V(p',s')-V(p',s)>0. 



§5 Bernoulli's equation 9 

Expanding this difference in powers of s'-s = gdsjdz, we obtain 

> 0. (4.1) 



/ 8V \ ds 
\ 8s Jp dz 



dz 
The formulae of thermodynamics give 

8V\ T I 8V 

p Cp \ 8 1 J p 
where c v is the specific heat at constant pressure. Both c p and T are positive, 



(v)-fO- 

\ 8s 1 v Cp \ 81 ) p 

it at 
so that we can write (4.1) as 



(^)^>0. (4.2) 

\8Tlpdz 

The majority of substances expand on heating, i.e. {8V]8T) V > 0. The 
condition that convection is absent then becomes 

ds/dz > 0, (4.3) 

i.e. the entropy must increase with height. 

From this we easily find the condition that must be satisfied by the 
temperature gradient dTjdz. Expanding the derivative dsjdz, we have 

ds _ / 8s \ dT /8s\ #_£p^_/j^\ ^ >0 
dz = \~8f)p~dz~ + \8pl T dz~ T 7 dz \8T/pdz > 
Finally, substituting from (3.4) dpjdz = -g\V, we obtain 

dT gT / 8V\ 

dz CpV \ 8T Jp 

Convection can occur if the temperature falls with increasing height and the 
magnitude of the temperature gradient exceeds (gTlc p V)(dV/8T)p. 
If we consider the equilibrium of a column of a perfect gas, then 

(T/V)(8V/8T)p = 1, 

and the condition for stable equilibrium is simply 

dTjdz > -gjcp. (4.5) 

§5. Bernoulli's equation 

The equations of fluid dynamics are much simplified in the case of steady 
flow. By steady flow we mean one in which the velocity is constant in time 
at any point occupied by fluid. In other words, v is a function of the co- 
ordinates only, so that dvj8t = 0. Equation (2.10) then reduces to 

|grada 2 -vxcurlv = -gradw. (5.1) 

We now introduce the concept of streamlines. These are lines such that 



10 Ideal Fluids 



§6 



the tangent to a streamline at any point gives the direction of the velocity 
at that point; they are determined by the following system of differential 
equations : 

dx dy dz 

VX Vy Vz ^ ' ' 

In steady flow the streamlines do not vary with time, and coincide with the 
paths of the fluid particles. In non-steady flow this coincidence no longer 
occurs: the tangents to the streamlines give the directions of the velocities 
of fluid particles at various points in space at a given instant, whereas the 
tangents to the paths give the directions of the velocities of given fluid 
particles at various times. 

We form the scalar product of equation (5.1) with the unit vector tangent 
to the streamline at each point; this unit vector is denoted by 1. The pro- 
jection of the gradient on any direction is, as we know, the derivative in that 
direction. Hence the projection of grad to is dtofdl. The vector vxcurl v 
is perpendicular to v, and its projection on the direction of 1 is therefore 
zero. 

Thus we obtain from equation (5.1) 

8 
-jfiv* + v>) = 0. 

It follows from this that \v 2 +w is constant along a streamline: 

%v 2 + w = constant. (5 # 3) 

In general the constant takes different values for different streamlines. 
Equation (5.3) is called Bernoulli's equation. 

If the flow takes place in a gravitational field, the acceleration g due to 
gravity must be added to the right-hand side of equation (5.1). Let us take 
the direction of gravity as the *-axis, with z increasing upwards. Then the 
cosine of the angle between the directions of g and 1 is equal to the derivative 
— dsr/d/, so that the projection of g on 1 is 

-gdzfdl. 

Accordingly, we now have 

8 
—(iv 2 + w+gz) = 0. 

Thus Bernoulli's equation states that along a streamline 

%v 2 +zo+gz = constant. (5.4) 

§6. The energy flux 

Let us choose some volume element fixed in space, and find how the 



§6 The energy flux H 

energy of the fluid contained in this volume element varies with time. 
The energy of unit volume of fluid is 

%pv 2 +pe, 

where the first term is the kinetic energy and the second the internal energy, 
€ being the internal energy per unit mass. The change in this energy is 
given by the partial derivative 

d 

—$pv z + pe). 
dt 

To calculate this quantity, we write 

d dp dv 

— (W) = i^ 2 ^ + pV-— , 

8f dt dt 

or, using the equation of continuity (1.2) and the equation of motion (2.3), 

o 

— (ipv 2 ) = -^2div(pv)-v.grad/>-pv-(v.grad)v. 

dt " 

In the last term we replace v • (v • grad)v by \v • grad v\ and grad/> by 
p grad zv-pT grad s (using the thermodynamic relation dw = Tds + (l/p)d/>), 
obtaining 

pi 

—Qpv 2 ) = -|a 2 div(pv)-pv.grad(^ 2 + «0+p:Zv.grad*. 
dt 

In order to transform the derivative d(pe)ldt, we use the thermodynamic 
relation 

de = Tds-pdV = Tds+(pl P 2 )dp. 
Since €+pjp = e+pV is simply the heat function w per unit mass, we find 

d[p € ) = edp + pde = zvdp+pTds, 
and so 

J^Z _ wJL + pT— = -zo div (pv)- pTv* grad s. 
dt dt dt 

Here we have also used the general adiabatic equation (2.6). 

Combining the above results, we find the change in the energy to be 



— Qpv^+pe) = -(^ 2 +w)div(/>v)-/>v.grad(^ 2 +w), 
dt 



or, finally, 



p* 

— (ipv 2 + pe) = -div[pv(|© 2 +w)]. (6.1) 

dt 



12 Ideal Fluids 



§7 



In order to see the meaning of this equation, let us integrate it over some 
volume : 

— J (ipv2+ P e)dV= - J div[pv$v2 + w)]dV, 

or, converting the volume integral on the right into a surface integral, 

— J (i P v2+pe)dV = - j>pv$v*+w).df. ^ 

The left-hand side is the rate of change of the energy of the fluid in some 
given volume. The right-hand side is therefore the amount of energy 
flowing out of this volume in unit time. Hence we see that the expression 

pv{\v* + w) ( 6<3 ) 

may be called the energy flux density vector. Its magnitude is the amount of 
energy passing in unit time through unit area perpendicular to the direction 
of the velocity. 

The expression (6.3) shows that any unit mass of fluid carries with it during 
its motion an amount of energy w + %v\ The fact that the heat function w 
appears here, and not the internal energy e, has a simple physical signifi- 
cance. Putting w = e+plp, we can write the flux of energy through a closed 
surface in the form 

- j>pv(±v 2 + e)-df- jpvdf. 

The first term is the energy (kinetic and internal) transported through the 
surface in unit time by the mass of fluid. The second term is the work done 
by pressure forces on the fluid within the surface. 

§7. The momentum flux 

We shall now give a similar series of arguments for the momentum of the 
fluid. The momentum of unit volume is pv. Let us determine its rate of 
change, 8(pv)/dt. We shall use tensor notation.f We have 

8 dvt 8p 

-(^) = /) __ + _^ 



a J J he La ? n S "f xes *' *' - take the v f Iues 1. 2, 3, corresponding to the components of vectors 
and tensors along the axes x, y, z respectively. We shall write sums of the type A • B = A.B,+ AB + 
a ~ ~f ^ * n the . form AfBt simply, omitting the summation sign. We shall use a similar pro- 
cedure in all products involving vectors or tensors: summation over the values 1, 2, 3 is always under- 
stood when a Latin suffix appears twice in any term. Such suffixes are sometimes called dummy 
suffixes In working with dummy suffixes it should be remembered that any pair of such suffixes may 
be replaced by any other like letters, since the notation used for suffixes that take all possible values 
obviously does not affect the value of the sum. 



§7 The momentum flux 13 

Using the equation of continuity (1.2) (with div(pv) written in the form 

d{pv k )jdx k ) 

dp d(pv k ) 

dt dx k ' 

and Euler's equation (2.3) in the form 

dv{ dvi 1 dp 

= — V k , 

dt dxjc p dxt 



we obtain 



d dvt dp d(pv k ) 
(pVi) = — pv k Vi— 

dV dx k dxt dx k 



= - — - —ipViV k ). 

dxi dx k 

We write the first term on the right in the formf 

dp dp 

—— = oi k - — , 

dxi dx k 



and finally obtain 



d dU ik 



where the tensor Ui k is defined as 

Ilta = p8 ik +pviv k . (7.2) 

This tensor is clearly symmetrical. 

To see the meaning of the tensor 11^, we integrate equation (7.1) over some 
volume : 

dt] H J dx k 

The integral on the right is transformed into a surface integral by Green's 
formula :% 

j (pv t dV= -j>U ik df k . (7.3) 



t 8,* denotes the unit tensor, i.e. the tensor with components which are unity for i ; = k and zero 
for »' 4= k. It is evident that h i]c A k = A it where A t is any vector. Similarly, if Am is a tensor of rank 
two, we have the relations BaAia = An, BaAa = A iit and so on. 

J The rule for transforming an integral over a closed surface into one over the volume bounded 
by that surface can be formulated as follows: the surface element d/ f must be replaced by the operator 
dV • 8/dxi, which is to be applied to the whole of the integrand. 



14 Ideal Fluids 



§8 



The left-hand side is the rate of change of the *th component of the 
momentum contained in the volume considered. The surface integral on 
the right is therefore the amount of momentum flowing out through the 
bounding surface in unit time. Consequently, U ik df k is the rth component 
of the momentum flowing through the surface element d/. If we write d/* 
in the form n k d/, where d/is the area of the surface element, and n is a unit 
vector along the outward normal, we find that Yl ik n k is the flux of the tth 
component of momentum through unit surface area. We may notice that, 
according to (7.2), Il ik n k = pn t + P ViV k n k . This expression can be written 
in vector form 

pn+ P v(vn). (7.4) 

Thus Tl ik is the jth component of the amount of momentum flowing in 
unit time through unit area perpendicular to the a^-axis. The tensor U ik 
is called the momentum flux density tensor. The energy flux is determined by 
a vector, energy being a scalar; the momentum flux, however, is determined 
by a tensor of rank two, the momentum itself being a vector. 

The vector (7.4) gives the momentum flux in the direction of n, i.e. 
through a surface perpendicular to n. In particular, taking the unit vector 
n to be directed parallel to the fluid velocity, we find that only the longitu- 
dinal component of momentum is transported in this direction, and its 
flux density is p + pv 2 . In a direction perpendicular to the velocity, only the 
transverse component (relative to v) of momentum is transported, its flux 
density being just p. 

§8. The conservation of circulation 

The integral 

T = jvdl 

taken along some closed contour, is called the velocity circulation round that 
contour. 

Let us consider a closed contour drawn in the fluid at some instant. 
We suppose it to be a "fluid contour", i.e. composed of the fluid particles 
that lie on it. In the course of time these particles move about, and the 
contour moves with them. Let us investigate what happens to the velocity 
circulation. In other words, let us calculate the time derivative 



Tt§ 



v-dl. 



We have written here the total derivative with respect to time, since we are 
seeking the change in the circulation round a "fluid contour" as it moves 
about, and not round a contour fixed in space. 
To avoid confusion, we shall temporarily denote differentiation with respect 



§8 The conservation of circulation 15 

to the co-ordinates by the symbol S, retaining the symbol d for differentia- 
tion with respect to time. Next, we notice that an element dl of the length 
of the contour can be written as the difference Sr between the radius vectors 
r of the points at the ends of the element. Thus we write the velocity cir- 
culation as § v • or. In differentiating this integral with respect to time, it 
must be borne in mind that not only the velocity but also the contour itself 
(i.e. its shape) changes. Hence, on taking the time differentiation under the 
integral sign, we must differentiate not only v but also Sr: 

d r f dv r dSr 

— * v«Sr = (b — «Sr + <b v— — . 
dtj 7 dt 7 dt 

Since the velocity v is just the time derivative of the radius vector r, 
we have 

v .^ = v.S— = v-Sv = S(i* 2 ). 
dt dt ^ 

The integral of a total differential along a closed contour, however, is zero. 
The second integral therefore vanishes, leaving 

d r w r dv 
— d) v«or = d) ——or. 
dt 7 7 dt 

It now remains to substitute for the acceleration dv[dt its expression 
from (2.9): 

dv/dt = — gradw. 
Using Stokes' formula, we then have 

since curl grad w = 0. Thus, going back to our previous notation, we 
findf 



or 



<j>vdl = constant. (8-1) 

We have therefore reached the conclusion that, in an ideal fluid, the velocity 
circulation round a closed "fluid" contour is constant in time {Kelvin's 
theorem or the law of conservation of circulation). 

It should be emphasised that this result has been obtained by using Euler's 
equation in the form (2.9), and therefore involves the assumption that the 



t This result remains valid in a uniform gravitational field, since in that case curl g = 0. 



16 Ideal Fluids §9 

flow is isentropic. The theorem does not hold for flows which are not 
isentropic.f 

§9. Potential flow 

From the law of conservation of circulation we can derive an important 
result. Let us at first suppose that the flow is steady, and consider a stream- 
line of which we know that to = curl v (the vorticity) is zero at some 
point. We draw an arbitrary infinitely small closed contour to encircle the 
streamline at that point. By Stokes' theorem, the velocity circulation round 
any infinitely small contour is equal to curl v . df, where df is the element of 
area enclosed by the contour. Since the contour at present under considera- 
tion is situated at a point where to = 0, the velocity circulation round it is 
zero. In the course of time, this contour moves with the fluid, but always 
remains infinitely small and always encircles the same streamline. Since 
the velocity circulation must remain constant, i.e. zero, it follows that o 
must be zero at every point on the streamline. 

Thus we reach the conclusion that, if at any point on a streamline to = 0, 
the same is true at all other points on that streamline. If the flow is not 
steady, the same result holds, except that instead of a streamline we must 
consider the path described in the course of time by some particular fluid 
particle; J we recall that in non-steady flow these paths do not in general 
coincide with the streamlines. 

At first sight it might seem possible to base on this result the following 
argument. Let us consider steady flow past some body. Let the incident 
flow be uniform at infinity; its velocity v is a constant, so that co s on all 
streamlines. Hence we conclude that to is zero along the whole of every 
streamline, i.e. in all space. 

A flow for which to = in all space is called a potential flow or irrotational 
flow, as opposed to rotational flow, in which the vorticity is not everywhere 
zero. Thus we should conclude that steady flow past any body, with a 
uniform incident flow at infinity, must be potential flow. 

Similarly, from the law of conservation of circulation, we might argue 
as follows. Let us suppose that at some instant we have potential flow 
throughout the volume of the fluid. Then the velocity circulation round any 
closed contour in the fluid is zero.ff By Kelvin's theorem, we could then 
conclude that this will hold at any future instant, i.e. we should find that, if 

t Mathematically, it is necessary that there should be a one-to-one relation between/) and p (which 
tor isentropic flow is s(p, p) - constant); then -(1/p) grad/> can be written as the gradient of some 
function, a result which is needed in deriving Kelvin's theorem. 

J To avoid misunderstanding, we may mention here that this result has no meaning in turbulent 
flow (cf. Chapter III). We may also remark that a non-zero vorticity may occur on a streamline after 
the passage of a shock wave. We shall see that this is because the flow is no longer isentropic, and the 
law of conservation of circulation cannot then be derived (§106). 

tt Here we suppose for simplicity that the fluid occupies a simply-connected region of space The 
same final result would be obtained for a multiply-connected region, but restrictions on the choice of 
contours would have to be made in the derivation. 



§9 



Potential flow 



17 



there is potential flow at some instant, then there is potential flow at all 
subsequent instants (in particular, any flow for which the fluid is initially 
at rest must be a potential flow). This is in accordance with the fact that, 
if to = 0, equation (2.11) is satisfied identically. 

In fact, however, all these conclusions are of only very limited validity. 
The reason is that the proof given above that to = all along a streamline 
is, strictly speaking, invalid for a line which lies in the surface of a solid 
body past which the flow takes place, since the presence of this surface makes 
it impossible to draw a closed contour in the fluid encircling such a stream- 
line. The equations of motion of an ideal fluid therefore admit solutions for 
which separation occurs at the surface of the body: the streamlines, having 
followed the surface for some distance, become separated from it at some 
point and continue into the fluid. The resulting flow pattern is characterised 
by the presence of a "surface of tangential discontinuity" proceeding from 
the body; on this surface the fluid velocity, which is everywhere tangential 
to the surface, has a discontinuity. In other words, at this surface one layer 
of fluid "slides" on another. Fig. 1 shows a surface of discontinuity which 
separates moving fluid from a region of stationary fluid behind the body. 




Fig. 1 

From a mathematical point of view, the discontinuity in the tangential velocity 
component corresponds to a surface on which the vorticity is non-zero. 

When such discontinuous flows are included, the solution of the equations 
of motion for an ideal fluid is not unique : besides continuous flow, they admit 
also an infinite number of solutions possessing surfaces of tangential dis- 
continuity starting from any prescribed line on the surface of the body 
past which the flow takes place. It should be emphasised, however, that 
none of these discontinuous solutions is physically significant, since tangen- 
tial discontinuities are wholly unstable, and therefore the flow would in fact 
become turbulent (see Chapter III). 

The actual physical problem of flow past a given body has, of course, a 
unique solution. The reason is that ideal fluids do not really exist; any 
actual fluid has a certain viscosity, however small. This viscosity may have 
practically no effect on the motion of most of the fluid, but, no matter how 
small it is, it will be important in a thin layer of fluid adjoining the body. 



18 Ideal Fluids §9 

The properties of the flow in this boundary layer decide the choice of one out 
of the infinity of solutions of the equations of motion for an ideal fluid. 
It is found that, in the general case of flow past bodies of arbitrary form, 
solutions with separation must be rejected; separation, if it occurred, would 
result in turbulence. 

In spite of what we have said above, the study of the solutions of the 
equations of motion for continuous steady potential flow past bodies is in 
some cases meaningful. Although, in the general case of flow past bodies of 
arbitrary form, the actual flow pattern bears almost no relation to the pattern 
of potential flow, for bodies of certain special ("streamlined"— §46) shapes 
the flow may differ very little from potential flow; more precisely, it will be 
potential flow except in a thin layer of fluid at the surface of the body and in 
a relatively narrow "wake" behind the body. 

Another important case of potential flow occurs for small oscillations of 
a body immersed in fluid. It is easy to show that, if the amplitude a of the 
oscillations is small compared with the linear dimension / of the body 
(a <^ /), the flow past the body will be potential flow. To show this, we esti- 
mate the order of magnitude of the various terms in Euler's equation 
dv/dt+(vgrad)v = — gradw. 

The velocity v changes markedly (by an amount of the same order as the 
velocity u of the oscillating body) over a distance of the order of the dimen- 
sion / of the body. Hence the derivatives of v with respect to the co-ordinates 
are of the order of u\l. The order of magnitude of v itself (at fairly small 
distances from the body) is determined by the magnitude of u. Thus we 
have (v • grad)v ~ u 2 /l. The derivative dvjdt is of the order of <ou, where 
o> is the frequency of the oscillations. Since w ~ uja, we have dvjdt ~ u 2 /a. 
It now follows from the inequality a <^l that the term (v • grad)v is small 
compared with dvjdt and can be neglected, so that the equation of motion 
of the fluid becomes dvjdt = -grad zo. Taking the curl of both sides, we 
obtain d(curl v)jdt = 0, whence curl v = constant. In oscillatory motion, 
however, the time average of the velocity is zero, and therefore curl v 
= constant implies that curl v = 0. Thus the motion of a fluid executing 
small oscillations is potential flow to a first approximation. 

We shall now obtain some general properties of potential flow. We first 
recall that the derivation of the law of conservation of circulation, and there- 
fore all its consequences, were based on the assumption that the flow is 
isentropic. If the flow is not isentropic, the law does not hold, and therefore, 
even if we have potential flow at some instant, the vorticity will in general 
be non-zero at subsequent instants. Thus only isentropic flow can in fact 
be potential flow. 
According to Stokes' theorem, 

<pv«dl = <j>curlv»df, 
where the integral on the right is taken over a surface bounded by the contour 



§9 Potential flow 19 

in question. Hence we see that, in potential flow, the velocity circulation 
round any closed contour is zero: 

£vdl = 0. (9.1) 

It follows from this that, in particular, closed streamlines cannot exist in 
potential flow.f For, since the direction of a streamline is at every point 
the direction of the velocity, the circulation along such a line can never be 
zero. 

In rotational motion the velocity circulation is not in general zero. In 
this case there may be closed streamlines, but it must be emphasised that the 
presence of closed streamlines is not a necessary property of rotational 
motion. 

Like any vector field having zero curl, the velocity in potential flow can 
be expressed as the gradient of some scalar. This scalar is called the velocity 
potential', we shall denote it by (f>: 

v = grad<£. (9.2) 

Writing Euler's equation in the form (2.10) 

0v/d*+|grada 2 -vxcurlv = -gradw 
and substituting v = grad <f>, we have 



grad I — + \v 2 +w I = 0, 



whence 

d<f>!dt+^v 2 +w =f(t), (9.3) 

where f(t) is an arbitrary function of time. This equation is a first integral 
of the equations of potential flow. The function /(*) in equation (9.3) can 
be put equal to zero without loss of generality. For, since the velocity is 
the space derivative of <j>, we can add to <f> any function of the time; replacing 
by <f> + $f(t)dt, we obtain zero on the right-hand side of (9.3). 

For steady flow we have (taking the potential cf> to be independent of time) 
dcf>jdt = 0, f{t) = constant, and (9.3) becomes Bernoulli's equation: 

\v 2 +w = constant. (9.4) 

It must be emphasised here that there is an important difference between the 
Bernoulli's equation for potential flow and that for other flows. In the 
general case, the "constant" on the right-hand side is a constant along any 
given streamline, but is different for different streamlines. In potential flow, 



t This result, like (9.1), may not be valid for motion in a multiply-connected region of space. 
In potential flow in such a region, the velocity circulation may be non-zero if the closed contour 
round which it is taken cannot be contracted to a point without crossing the boundaries of the region. 



20 Ideal Fluids 



§10 



however, it is constant throughout the fluid. This enhances the importance 
of Bernoulli's equation in the study of potential flow. 

§10. Incompressible fluids 

In a great many cases of the flow of liquids (and also of gases), their 
density may be supposed invariable, i.e. constant throughout the volume of 
the fluid and throughout its motion. In other words, there is no noticeable 
compression or expansion of the fluid in such cases. We then speak of 
incompressible flow. 

The general equations of fluid dynamics are much simplified for an 
incompressible fluid. Euler's equation, it is true, is unchanged if we put 
p = constant, except that p can be taken under the gradient operator in 
equation (2.4) : 



— + (v-grad)v = -grad/ - j +g. 



(10.1) 



The equation of continuity, on the other hand, takes for constant p the 
simple form 

div v = 0. (10.2) 

Since the density is no longer an unknown function as it was in the general 
case, the fundamental system of equations in fluid dynamics for an incom- 
pressible fluid can be taken to be equations involving the velocity only 
These may be the equation of continuity (10.2) and equation (2.11): 

8 
— (curl v) = curl <Vx curl v). (10.3) 

Bernoulli's equation can be written in a simpler form for an incompressible 
fluid. Equation (10.1) differs from the general Euler's equation (2.9) in that 
it has grad {pip) in place of grad w. Hence we can write down Bernoulli's 
equation immediately by simply replacing the heat function in (5.4) by pjp: 

%v 2 +p/p+gz = constant. (10.4) 

For an incompressible fluid, we can also write pjp in place of w in the 
expression (6.3) for the energy flux, which then becomes 



{v + >\ 



pv\&*+-y (io.5) 

For we have, from a well-known thermodynamic relation, the expression 
de = Tds-pdV for the change in internal energy; for s = constant and 
V ~ \Jp = constant, de = 0, i.e. e = constant. Since constant terms in 
the energy do not matter, we can omit e in zv = e+pjp. 



§10 



Incompressible fluids 



21 



The equations are particularly simple for potential flow of an incom- 
pressible fluid. Equation (10.3) is satisfied identically if curl v = 0. Equa- 
tion (10.2), with the substitution v = grad <f>, becomes 



A<£ = 0, 



(10.6) 



i.e. Laplace's equation^ for the potential <f>. This equation must be supple- 
mented by boundary conditions at the surfaces where the fluid meets solid 
bodies. At fixed solid surfaces, the fluid velocity component v n normal to 
the surface must be zero, whilst for moving surfaces it must be equal to the 
normal component of the velocity of the surface (a given function of time). 
The velocity v n , however, is equal to the normal derivative of the potential 
<f> : v n = 8<f>ldn. Thus the general boundary conditions are that dj>\dn is 
a given function of co-ordinates and time at the boundaries. 

For potential flow, the velocity is related to the pressure by equation (9.3). 
In an incompressible fluid, we can replace zo in this equation by pjp : 

a+iat+w+p/p = /(')• (10.7) 

We may notice here the following important property of potential flow of 
an incompressible fluid. Suppose that some solid body is moving through 
the fluid. If the result is potential flow, it depends at any instant only on 
the velocity of the moving body at that instant, and not, for example, on its 
acceleration. For equation (10.6) does not explicitly contain the time, which 
enters the solution only through the boundary conditions, and these contain 
only the velocity of the moving body. 




Fig. 2 



From Bernoulli's equation, %v 2 +plp = constant, we see that, in steady 
flow of an incompressible fluid (not in a gravitational field), the greatest 
pressure occurs at points where the velocity is zero. Such a point usually 
occurs on the surface of a body past which the fluid is moving (at the point 
O in Fig. 2), and is called a stagnation point. If u is the velocity of the 



t The velocity potential was first introduced by Euler, who obtained an equation of the form 
(10.6) for it; this form later became known as Laplace's equation. 



22 Ideal Fluids §10 

incident current (i.e. the fluid velocity at infinity), and po the pressure at 
infinity, the pressure at the stagnation point is 

pmzx = po + lpuK (10.8) 

If the velocity distribution in a moving fluid depends on only two co- 
ordinates (x and y, say), and the velocity is everywhere parallel to the 
ry-plane, the flow is said to be two-dimensional or plane flow. To solve 
problems of two-dimensional flow of an incompressible fluid, it is sometimes 
convenient to express the velocity in terms of what is called the stream 
function. From the equation of continuity divv = dvxjdx+dvy/dy = we 
see that the velocity components can be written as the derivatives 

v x = difjjdy, v y =- di/jjdx (10.9) 

of some function ip(x, y), called the stream function. The equation of con- 
tinuity is then satisfied automatically. The equation that must be satisfied 
by the stream function is obtained by substituting (10.9) in equation (10.3). 
We then obtain 

d 8ib 8 8ib 8 

* A *- a 7a7 A * + ^ A * = °- (1<U0) 

If we know the stream function we can immediately determine the form of 
the streamlines for steady flow. For the differential equation of the stream- 
lines (in two-dimensional flow) is dxjv x = dyjv y or v y dx — v x dy = ; 
it expresses the fact that the direction of the tangent to a streamline is the 
direction of the velocity. Substituting (10.9), we have 

difj 8i[i 

— dx H dy = d^r = 0, 

dx dy 

whence j/t == constant. Thus the streamlines are the family of curves obtained 
by putting the stream function \p{x, y) equal to an arbitrary constant. 

If we draw a curve between two points A and B in the ary-plane, the mass 
flux Q across this curve is given by the difference in the values of the stream 
function at these two points, regardless of the shape of the curve. For, if 
v n is the component of the velocity normal to the curve at any point, we have 

B B B 

Q = p fv n dl = p j> (-v y dx+v x dy) = p \ dip, 

A A A 

or 

Q = Mb-Ia). (10.11) 

There are powerful methods of solving problems of two-dimensional poten- 
tial flow of an incompressible fluid past bodies of various profiles, involving 



§10 Incompressible fluids 23 

the application of the theory of functions of a complex variable.f The basis 
of these methods is as follows. The potential and the stream function are 
related to the velocity components by 

v x = dcf>/dx = difi/dy, v y = 8<f>jdy = - dip/dx. 

These relations between the derivatives of <j> and ijj, however, are the same, 
mathematically, as the well-known Cauchy-Riemann conditions for a complex 
expression 

w = <f> + irfi (10.12) 

to be an analytic function of the complex argument z = x+iy. This means 
that the function w(z) has at every point a well-defined derivative 

dw d<f> dift 

— — = \- i — = v x —Wy. (10.13) 

dz 8x 8x y v ' 

The function to is called the complex potential, and dwfdz the complex velocity. 
The modulus and argument of the latter give the magnitude v of the velocity 
and the angle 6 between the direction of the velocity and that of the #-axis : 

dzvjdz = ve-*°. (10.14) 

At a solid surface past which the flow takes place, the velocity must be 
along the tangent. That is, the profile contour of the surface must be a 
streamline, i.e. ^ = constant along it; the constant may be taken as zero, 
and then the problem of flow past a given contour reduces to the deter- 
mination of an analytic function tv(z) which takes real values on the contour. 
The statement of the problem is more involved when the fluid has a free 
surface ; an example is found in Problem 9. 

The integral of an analytic function round any closed contour C is well 
known to be equal to 2-rri times the sum of the residues of the function at its 
simple poles inside C; hence 

<J> w'dz = 2rri ^ Afo 
k 

where Ak are the residues of the complex velocity. We also have 

<j> w' dz = <x> (v x — iv y )(dx + idy) 

= <j> (v x dx + v y dy) + ij> (v x dy — v y dx). 



f A more detailed account of these methods and their various applications is given by N. E. Kochin, 
I. A. Kibel' and N. V. Roze, Theoretical Hydromechanics (Teoreticheskaya gidromekhanika), Part 1, 
4th ed., Moscow 1948; L. I. Sedov, Two-dimensional Problems of Hydrodynamics and Aerodynamics 
(Ploskie zadachi gidrodinamxki i a'erodinamiki), Moscow 1950. 



24 Ideal Fluids §10 

The real part of this expression is just the velocity circulation V round 
the contour C. The imaginary part, multiplied by p, is the mass flux across 
C; if there are no sources of fluid within the contour, this flux is zero and 
we then have simply 

r = 2in^A k ; (10.15) 



all the residues Ajc are in this case purely imaginary. 

Finally, let us consider the conditions under which the fluid may be 
regarded as incompressible. When the pressure changes adiabatically by 
Ap, the density changes by Ap = {dpjdp) 8 Ap. According to Bernoulli's 
equation, however, Ap is of the order of pv 2 in steady flow. Thus Ap ~ 
{dpjdp) s pv 2 . We shall show in §63 that the derivative (8pjdp) s is the square 
of the velocity c of sound in the fluid, so that Ap ~ pv 2 jc 2 . The fluid may be 
regarded as incompressible if Ap/p <^ 1. We see that a necessary condition 
for this is that the fluid velocity should be small compared with that of 
sound : 

v <£ c. (10.16) 

However, this condition is sufficient only in steady flow. In non-steady 
flow, a further condition must be fulfilled. Let t and / be a time and a length 
of the order of the times and distances over which the fluid velocity undergoes 
significant changes. If the terms dvjdt and (l//>) gradp in Euler's equation 
are comparable, we find, in order of magnitude, vjr ~ Apjlp or Ap ~ Ipvjr, 
and the corresponding change in p is Ap ~ Ipvjrc 2 . Now comparing the terms 
dp/dt and p div v in the equation of continuity, we find that the derivative 
dpjdt may be neglected (i.e. we may suppose p constant) if Apjr <^ pvjl, 
or 

t > Ijc. (10.17) 

If the conditions (10.16) and (10.17) are both fulfilled, the fluid may be 
regarded as incompressible. The condition (10.17) has an obvious meaning: 
the time Ijc taken by a sound signal to traverse the distance / must be small 
compared with the time t during which the flow changes appreciably, so 
that the propagation of interactions in the fluid may be regarded as instan- 
taneous. 

PROBLEMS 

Problem 1. Determine the shape of the surface of an incompressible fluid subject to a 
gravitational field, contained in a cylindrical vessel which rotates about its (vertical) axis with 
a constant angular velocity fi. 

Solution. Let us take the axis of the cylinder as the ar-axis. Then vx = — yCl, v y = *Q, 



§10 Incompressible fluids 25 

vz = 0. The equation of continuity is satisfied identically, and Euler's equation (10.1) 
gives 

p ox p ay p oz 

The general integral of these equations is 

p/p = %Q. 2 (x 2 +y 2 )—gz+ constant. 

At the free surface p = constant, so that the surface is a paraboloid: 

z = & 2 (x 2 +y 2 )/g t 

the origin being taken at the lowest point of the surface. 

Problem 2. A sphere, of radius R, moves with velocity u in an incompressible ideal fluid. 
Determine the potential flow of the fluid past the sphere. 

Solution. The fluid velocity must vanish at infinity. The solutions of Laplace's equation 
A <f> = which vanish at infinity are well known to be ljr and the derivatives, of various orders, 
of 1/r with respect to the co-ordinates (the origin is taken at the centre of the sphere). On 
account of the complete symmetry of the sphere, only one constant vector, the velocity u, 
can appear in the solution, and, on account of the linearity of both Laplace's equation and 
the boundary condition, <f> must involve u linearly. The only scalar which can be formed 
from u and the derivatives of 1/r is the scalar product u • grad(l/r). We therefore seek ^ 
in the form 

<f> = A.grad(l/r) = -(A.n)/r2, 

where n is a unit vector in the direction of r. The constant A is determined from the condition 
that the normal components of the velocities v and u must be equal at the surface at the 
sphere, i.e. vn = u*n for r = R. This condition gives A = iuR 3 , so that 

The pressure distribution is given by equation (10.7) : 

P = po-^pv 2 -p8<f>Jdt, 

where p is the pressure at infinity. To calculate the derivative 8<f>[8t, we must bear in mind 
that the origin (which we have taken at the centre of the sphere) moves with velocity u. 
Hence 

d<f>/dt = (d(f>/du)-u-wgrad(f). 

The pressure distribution over the surface of the sphere is given by the formula 

P = po+$pu 2 (9 cos 2 d-5)+ipRn'du/dt, 

where 6 is the angle between n and u. 

Problem 3. The same as Problem 2, but for an infinite cylinder moving perpendicular to 
its axis.f 



t The solution of the more general problems of potential flow past an ellipsoid and an elliptical 
cylinder may be found in: N. E. Kochin, I. A. Kibel' and N. V. Roze, Theoretical Hydromechanics 
(Teoreticheskaya gidromekhanika), Part 1, 4th ed., pp. 265 and 355, Moscow 1948; H. Lamb, Hydro- 
dynamics, 6th ed., §§103-116, Cambridge 1932. 



26 Ideal Fluids §10 

Solution. The flow is independent of the axial co-ordinate, so that we have to solve 
Laplace's equation in two dimensions. The solutions which vanish at infinity are the first 
and higher derivatives of log r with respect to the co-ordinates, where r is the radius vector 
perpendicular to the axis of the cylinder. We seek a solution in the form 

<£ = A«gradlogr = A»n/r, 

and from the boundary conditions we obtain A = — B?u, so that 

R 2 R 2 

9= u«n, v = -~[2n(u.n)-u]. 

The pressure at the surface of the cylinder is given by the formula 

P = A>+|pw 2 (4 cos 2 d-3)+ p Rn-duJdt. 

Problem 4. Determine the potential flow of an incompressible ideal fluid in an ellipsoidal 
vessel rotating about a principal axis with angular velocity Q, and determine the total angular 
momentum of the fluid. 

Solution. We take Cartesian co-ordinates x, y, z along the axes of the ellipsoid at a given 
instant, the z-axis being the axis of rotation. The velocity of points in the vessel is 

u = SI x r, 

so that the boundary condition v n = d<$>\dn = «„ is 

d$\dn = Q(xn y —yn x ), 

or, using the equation of the ellipsoid x 2 /a 2 +y 2 Jb 2 +z 2 /c 2 = 1, 

x d(f> 

a 2 8x b 2 dy 
The solution of Laplace's equation which satisfies this boundary condition is 

a 2 -b 2 
* = Q a^b 2Xy ' (1) 

The angular momentum of the fluid in the vessel is 

M = p J (xv y -yv x )dV. 

Integrating over the volume V of the ellipsoid, we have 

QpV (cfi-b 2 ) 2 



y d 9 z 8 9 / 1 1 \ 

b 2 dy c 2 8z J \ b 2 a 2 J 



M = 



a 2 +b 2 



Formula (1) gives the absolute motion of the fluid relative to the instantaneous position 
of the axes x, y, z which are fixed to the rotating vessel. The motion relative to the vessel 
(i.e. relative to a rotating system of co-ordinates *, y, z) is found by subtracting the velocity 
SiXr from the absolute velocity; denoting the relative velocity of the fluid by v', we have 

d 9 ^ 2Q.a 2 , 2Q&2 

v x = _ + y a = v, v ' = - v v' z = o. 

dx a 2 + b 2 a 2 + b 2 

The paths of the relative motion are found by integrating the equations x = v' x , y = v' y , 
and are the ellipses x s /a z +y 2 /b 2 = constant, which are similar to the boundary ellipse. 



§10 Incompressible fluids 27 

Problem 5. Determine the flow near a stagnation point (Fig. 2). 

Solution. A small part of the surface of the body near the stagnation point may be 
regarded as plane. Let us take it as the ary-plane. Expanding <f> for *, y, z small, we have 
as far as the second-order terms 

<j> = ax+by + cz+Ax 2 +By 2 + Cz 2 +Dxy + Eyz+Fzx; 

a constant term in <f> is immaterial. The constant coefficients are determined so that <f> satisfies 
the equation A^ = and the boundary conditions v t = d<f>/dz = for z — and all x, y, 
d<f>Jdx = d<f>/dy — 0fotx=y = z = (the stagnation point). This gives a = b = c = 0; 
C = —A —B, E = F = 0. The term Dxy can always be removed by an appropriate rotation 
of the * and y axes. We then have 

<f> = Ax 2 + By 2 -(A + B)z 2 . (1) 

If the flow is axially symmetrical about the s-axis (symmetrical flow past a solid of revo- 
lution), we must have A = B, so that 

<f> = A{x 2 +y 2 -2z 2 ). 

The velocity components are v x = 2Ax, v v = 2Ay, v z =■ —\Az. The streamlines are given 
by equations (5.2), from which we find x 2 z = c lt y 2 z = c it i.e. the streamlines are cubical 
hyperbolae. 

If the flow is uniform in the y-direction (e.g. flow in the #-direction past a cylinder with 
its axis in the y-direction), we must have B = in (1), so that 

<f> = A{x 2 -z*). 
The streamlines are the hyperbolae xz = constant. 

Problem 6. Determine the potential flow near an angle formed by two intersecting 
planes. 

Solution. Let us take polar co-ordinates r, 6 in the cross-sectional plane (perpendicular 
to the line of intersection), with the origin at the vertex of the angle ; 6 is measured from one 
of the arms of the angle. Let the angle be a radians ; for a < it the flow takes place within 
the angle, for a > v outside it. The boundary condition that the normal velocity component 
vanishes means that 8<f>l8B = for 6 = and = a. The solution of Laplace's equation 
satisfying these conditions can be written! 

<f> = Ar n cos nd, n — ir/a, 

so that 

v r = nAr 11 - 1 cos nd, v d = — nAr n smnd. 

For « < 1 (flow outside an angle; Fig. 3), v r becomes infinite as l/r 1_n at the origin. For 
n > 1 (flow inside an angle; Fig. 4), v T becomes zero for r = 0. 

The stream function, which gives the form of the streamlines, is tjt = Ar n sin nd. The 
expressions obtained for <j> and ^ are the real and imaginary parts of the complex potential 
to = Az n . 

Problem 7. A spherical hole of radius a is suddenly formed in an incompressible fluid 
filling all space. Determine the time taken for the hole to be filled with fluid (Rayleigh 
1917). 

Solution. The flow after the formation of the hole will be spherically symmetrical, the 



t We take the solution which involves the lowest positive power of r, since r is small. 



28 



Ideal Fluids 



§10 



velocity at every point being directed to the centre of the hole. For the radial velocity 
»,s»<0we have Euler's equation in spherical polar co-ordinates : 



dv dv 1 dp 

— + v— = -. 

dt dr p dr 



(1) 



The equation of continuity gives 



rh) = F(t), 



(2) 

where F(t) is an arbitrary function of time; this equation expresses the fact that, since the 
fluid is incompressible, the volume flowing through any spherical surface is independent of 
the radius of that surface. 




Fig. 3 




Fig. 4 



Substituting v from (2) in (1), we have 



F'(t) dv 1 dp 

— — + v — = 



8r 



p 8r 



Integrating this equation over r from the instantaneous radius R = R(t) < a of the hole to 
infinity, we obtain 



F'(t) po 

R P 



(3) 



where V = dR(t)!dt is the rate of change of the radius of the hole, and p is the pressure at 



§10 Incompressible fluids 29 

infinity; the fluid velocity at infinity is zero, and so is the pressure at the surface of the hole. 
From equation (2) for points on the surface of the hole we find 

F{i) = RHt)V{t\ 

and, substituting this expression for F(t) in (3), we obtain the equation 

3F2 dF2 p 

Integrating with the boundary condition V = for R — a (the fluid being initially at rest), 
we have 



dR _ 

Hence we have for the required total time for the hole to be filled 



dt V [_ 3p \R3 /. 



/ 3 P r dR 

T ~ J 2^} ^/[{ajRf-\] 

This integral reduces to a beta function, and we have finally 

V 2p Q r(i/3) V po 

Problem 8. A sphere immersed in an incompressible fluid expands according to a given 
law R = R(t). Determine the fluid pressure at the surface of the sphere. 

Solution. Let the required pressure be P(t). Calculations exactly similar to those of 
Problem 7, except that the pressure at r = R is P(t) and not zero, give instead of (3) the 
equation 

R * p p 

and accordingly instead of (4) the equation 

P 2 dR' 

Bearing in mind the fact that V = dR/dt, we can write the expression for P(t) in the form 



i rd2(i?2) /cLR\2-| 



Problem 9. Determine the form of a jet emerging from an infinitely long slit in a plane 
wall. 

Solution. Let the wall be along the *-axis in the xy-plane, and the aperture be the 
segment — \a < x < \a of that axis, the fluid occupying the half-plane y > 0. Far from the 
wall [y -*■ co) the fluid velocity is zero, and the pressure is p , say. 

At the free surface of the jet (BC and B'C in Fig. 5a) the pressure p = 0, while the velocity 



30 



Ideal Fluids 



§10 



takes the constant value v x = V(2po/p)> by Bernoulli's equation. The wall lines are stream- 
lines, and continue into the free boundary of the jet. Let ip be zero on the line ABC; then, 
on the line A'B'C, tft = —Q/p, where Q = pa^ is the rate at which the fluid emerges in 
the jet (a lt v x being the jet width and velocity at infinity). The potential <f> varies from — oo 
to + oo both along ABC and along A'B'C; let <f> be zero at B and B'. Then, in the plane of 
the complex variable w, the region of flow is an infinite strip of width Qjp (Fig. 5b). (The 
points in Fig. 5b, c, d are named to correspond with those in Fig. 5a.) 




c,c 



8 



(c) 



® 



->o 3 £ 



I* ® 



© 



C 
(b) 



l-Q/p 



\B 



-<£ / B' Cjc' B A 

=*1 | "l r 



(d) 



Fig. 5 



We introduce a new complex variable, the logarithm of the complex velocity: 
f 1 da; "] v± 



(1) 



here v 1 e iin is the complex velocity of the jet at infinity. On A'B' we have 6 = 0; on AB, 
9 ~ —t; on BC and B'C, v = v u while at infinity in the jet = \ir. In the plane of the 
complex variable £, therefore, the region of flow is a semi-infinite strip of width n in the 
right half-plane (Fig. 5c). If we can now find a conformal transformation which carries the 
strip in the to-plane into the half-strip in the £-plane (with the points corresponding as in 
Fig. 5), we shall have determined w as a function of dzo/ds, and zv can then be found by a 
simple quadrature. 

In order to find the desired transformation, we introduce one further auxiliary complex 
variable, u, such that the region of flow in the M-pIane is the upper half-plane, the points 
B and B' corresponding to u = ±1, the points C and C" to u — 0, and the infinitely distant 
points A and A' to u = ± oo (Fig. 5d). The dependence of v) on this auxiliary variable is 
given by the conformal transformation which carries the upper half of the u-plane into the 
strip in the w-plane. With the above correspondence of points, this transformation is 



to = lOgtt. 

fm 



(2) 



In order to find the dependence of £ on u, we have to find a conformal transformation of the 
half-strip in the £-plane into the upper half of the u-plane. Regarding this half-strip as a 



§11 The drag force in potential flow past a body 31 

triangle with one vertex at infinity, we can find the desired transformation by means of the 
well-known Schwarz-Christoffel formula; it is 

£ = — *sin -1 #. (3) 

Formulae (2) and (3) give the solution of the problem, since they furnish the dependence of 
dzo/dz on to in parametric form. 

Let us now determine the form of the jet. On BC we have to — <f>, £ = i($n+0), while u 
varies from 1 to 0. From (2) and (3) we obtain 

<f> = --^log(-cos0), (4) 

prr 
and from (1) we have 

d(f}jdz = vie- i0 , 
or 

d* = dx + i dy = — e^ d<f> = -V tanfl dd, 

Vl IT 

whence we find, by integration with the conditions y — 0,x = ia for 9 — —n, the form of the 
jet, expressed parametrically. In particular, the compression of the jet is aja = 7r/(2+w) 
= 0-61. 

§11. The drag force in potential flow past a body 

Let us consider the problem of potential flow of an incompressible ideal 
fluid past some solid body. This problem is, of course, completely equivalent 
to that of the motion of a fluid when the same body moves through it. To 
obtain the latter case from the former, we need only change to a system of 
co-ordinates in which the fluid is at rest at infinity. We shall, in fact, say in 
what follows that the body is moving through the fluid. 

Let us determine the nature of the fluid velocity distribution at great 
distances from the moving body. The potential flow of an incompressible 
fluid satisfies Laplace's equation, /\<f> = 0. We have to consider solutions 
of this equation which vanish at infinity, since the fluid is at rest there. 
We take the origin somewhere inside the moving body; the co-ordinate 
system moves with the body, but we shall consider the fluid velocity distri- 
bution at a particular instant. As we know, Laplace's equation has a solution 
1/r, where r is the distance from the origin. The gradient and higher space 
derivatives of 1/r are also solutions. All these solutions, and any linear 
combination of them, vanish at infinity. Hence the general form of the 
required solution of Laplace's equation at great distances from the body is 

a 1 

<f> = h A«grad- + ... , 

r r 

where a and A are independent of the co-ordinates; the omitted terms con- 
tain higher-order derivatives of 1/r. It is easy to see that the constant a 
must be zero. For the potential <f> = —ajr gives a velocity 

v = — grad(a/r) = ar/r s . 

Let us calculate the corresponding mass flux through some closed surface, 



32 Ideal Fluids §11 

say a sphere of radius R. On this surface the velocity is constant and equal 
to a/R 2 ; the total flux through it is therefore p{a\R 2 )^R 2 = Am pa. But the 
flux of an incompressible fluid through any closed surface must, of course, 
be zero. Hence we conclude that a = 0. 

Thus <j> contains terms of order \jr 2 and higher. Since we are seeking the 
velocity at large distances, the terms of higher order may be neglected, and 
we have 

<f> = A-grad(l/r) = -A-n/r 2 , (11.1) 

and the velocity v = grad <f> is 

v = (A-grad) grad- = -^ '- , (11.2) 

r r 3 

where n is a unit vector in the direction of r. We see that at large distances 
the velocity diminishes as l/r3. The vector A depends on the actual shape 
and velocity of the body, and can be determined only by solving completely 
the equation A<f> = at all distances, taking into account the appropriate 
boundary conditions at the surface of the moving body. 

The vector A which appears in (11.2) is related in a definite manner to 
the total momentum and energy of the fluid in its motion past the body. 
The total kinetic energy of the fluid (the internal energy of an incompressible 
fluid is constant) is E = % fpv 2 dV, where the integration is taken over all 
space outside the body. We take a region of space V bounded by a sphere of 
large radius R, whose centre is at the origin, and first integrate only over 
V, later letting R tend to infinity. We have identically 

jv 2 dV = ju 2 dV+ j (v+u).(v-u)dV, 

where u is the velocity of the body. Since u is independent of the co-ordinates, 
the first integral on the right is simply u 2 (V- V ), where V is the volume of 
the body. In the second integral, we write the sum v+uas grad (<f> + u . r) ; 
using the facts that div v = (equation of continuity) and div u = 0, we 
have 

j* v 2 dV = u\V- Vo)+ j div [(<£ + u-r)(v- u)]dF. 

The second integral is now transformed into an integral over the surface S 
of the sphere and the surface So of the body : 

jv 2 dV = u 2 (V-V )+ j> (^+u.r)(v-u).df. 

s+s a 

On the surface of the body, the normal components of v and u are equal by 
virtue of the boundary conditions ; since the vector df is along the normal 



§11 The drag force in potential flow past a body 33 

to the surface, it is clear that the integral over So vanishes identically. On 
the remote surface S we substitute the expressions (11.1), (11.2) for <f> and v, 
and neglect terms which vanish as R -> oo. Writing the surface element 
on the sphere S in the form df = nR 2 do, where do is an element of solid angle, 
we obtain 

jv*dV = u?(±ttB?-Vo)+ J" [3(A.n)(u.n)-(u.n)2fl3]do. 

Finally, effecting the integration! and multiplying by \p> we obtain the 
following expression for the total energy of the fluid : 

E = £ p (4ttA.u- V u*). (11.3) 

As has been mentioned already, the exact calculation of the vector A 
requires a complete solution of the equation /\<f> = 0» taking into account the 
particular boundary conditions at the surface of the body. However, the 
general nature of the dependence of A on the velocity u of the body can be 
found directly from the facts that the equation is linear in <f>, and the boundary 
conditions are linear in both <f> and u. It follows from this that A must be a 
linear function of the components of u. The energy E given by formula 
(11.3) is therefore a quadratic function of the components of u, and can be 
written in the form 

E = \m ik UiU k , (11.4) 

where m^ is some constant symmetrical tensor, whose components can be 
calculated from those of A; it is called the induced-mass tensor. 

Knowing the energy E, we can obtain an expression for the total momentum 
P of the fluid. To do so, we notice that infinitesimal changes in E and P 
are related by J dE = u • dP; it follows from this that, if E is expressed in 



t The integration over o is equivalent to averaging the integrand over all directions of the vector 
n and multiplying by 4w. To average expressions of t he ty pe (A • n)(B • n) = ^4,« tJ BfcWfc, where A, B 
are constant vectors, we notice that the mean values tiittje form a symmetrical tensor, which can be 
expressed in terms of the unit tensor S«: w,w* = aS,*. Contracting with respect to the suffixes i 
and k, and remembering that «*»!< = 1, we find that a = \. Hence 



(A-nXB.n) = VfaAtB* = £A- B. 

X For, let the body be accelerated by some external force F. The momentum of the fluid will thereby 
be increased; let it increase by dP during a time df. This increase is related to the force by dP = F df, 
and on scalar multiplication by the velocity u we have u • dP = F • u df, i.e. the work done by the 
force F acting through the distance u df , which in turn must be equal to the increase dE in the energy 
of the fluid. 

It should be noticed that it would not be possible to calculate the momentum directly as the integral 
pv dV over the whole volume of the fluid. The reason is that this integral, with the velocity v 
distributed in accordance with (11.2), diverges, in the sense that the result of the integration, though 
finite, depends on how the integral is taken: on effecting the integration over a large region, whose 
dimensions subsequently tend to infinity, we obtain a value depending on the shape of the region 
(sphere, cylinder, etc.). The method of calculating the momentum which we use here, starting from 
the relation u • dP = dE, leads to a completely definite final result, given by formula (11.6), which 
certainly satisfies the physical relation between the rate of change of the momentum and the forces 
acting on the body. 



34 Ideal Fluids §n 

the form (11.4), the components of P must be 

Pi = m ik u k . (H.5) 

Finally, a comparison of formulae (11.3), (11.4) and (11.5) shows that P 
is given in terms of A by 

P = 4tt P A- p Vo\i. (11.6) 

It must be noticed that the total momentum of the fluid is a perfectly definite 
finite quantity. 

The momentum transmitted to the fluid by the body in unit time is dP/d*. 
With the opposite sign it evidently gives the reaction F of the fluid, i.e. the 
force acting on the body : 

F = -dP/d*. (H.7) 

The component of F parallel to the velocity of the body is called the drag 
force, and the perpendicular component is called the lift force. 

If it were possible to have potential flow past a body moving uniformly 
in an ideal fluid, we should have P = constant, since u = constant, and so 
F = 0. That is, there would be no drag and no lift; the pressure forces 
exerted on the body by the fluid would balance out (a result known as 
d'Alemberfs paradox). The origin of this paradox is most clearly seen by 
considering the drag. The presence of a drag force in uniform motion of a 
body would mean that, to maintain the motion, work must be continually 
done by some external force, this work being either dissipated in the fluid or 
converted into kinetic energy of the fluid, and the result being a continual 
flow of energy to infinity in the fluid. There is, however, by definition 
no dissipation of energy in an ideal fluid, and the velocity of the fluid set in 
motion by the body diminishes so rapidly with increasing distance from the 
body that there can be no flow of energy to infinity. 

However, it must be emphasised that all these arguments relate only to 
the motion of a body in an infinite volume of fluid. If, for example, the 
fluid has a free surface, a body moving uniformly parallel to this surface will 
experience a drag. The appearance of this force (called wave drag) is due to 
the occurrence of a system of waves propagated on the free surface, which 
continually remove energy to infinity. 

Suppose that a body is executing an oscillatory motion under the action 
of an external force f. When the conditions discussed in §10 are fulfilled, 
the fluid surrounding the body moves in a potential flow, and we can use the 
relations previously obtained to derive the equations of motion of the body. 
The force f must be equal to the time derivative of the total momentum of 
the system, and the total momentum is the sum of the momentum Mvl 
of the body (M being the mass of the body) and the momentum P of the fluid : 

Mdu/d*+dP/d* = f. 



§1 1 The drag force in potential flow past a body 35 

Using (11.5), we then obtain 

M dui/dt + mac dujc/dt = /*, 
which can also be written 

^(M8 ik + m ik )=f i . (11.8) 

at 

This is the equation of motion of a body immersed in an ideal fluid. 

Let us now consider what is in some ways the converse problem. Suppose 
that the fluid executes some oscillatory motion on account of some cause 
external to the body. This motion will set the body in motion also.f We 
shall derive the equation of motion of the body. 

We assume that the velocity of the fluid varies only slightly over distances 
of the order of the dimension of the body. Let v be what the fluid velocity 
at the position of the body would be if the body were absent; that is, v is the 
velocity of the unperturbed flow. According to the above assumption, v 
may be supposed constant throughout the volume occupied by the body. 
We denote the velocity of the body by u as before. 

The force which acts on the body and sets it in motion can be determined 
as follows. If the body were wholly carried along with the fluid (i.e. if 
v = u), the force acting on it would be the same as the force which would act 
on the liquid in the same volume if the body were absent. The momentum of 
this volume of fluid is pVo\, and therefore the force on it is pVo dv/d*. 
In reality, however, the body is not wholly carried along with the fluid; 
there is a motion of the body relative to the fluid, in consequence of which 
the fluid itself acquires some additional motion. The resulting additional 
momentum of the fluid is mi k {u k -v k ), since in (11.5) we must now replace u 
by the velocity u-v of the body relative to the fluid. The change in this 
momentum with time results in the appearance of an additional reaction 
force on the body of -m ik d{u k -v k )\dt. Thus the total force on the body is 

pVo— m ik —(u k -v k ). 

dt dt 

This force is to be equated to the time derivative of the body momentum. 
Thus we obtain the following equation of motion: 

d dvi d 

—(Mui) = pV — m ik —(u k -v k ). 

dt dt dt 

Integrating both sides with respect to time, we have 

Mui = pVoVi-m ik (u k -v k ), 

or 

(MS ik + m ik )u k = (m ik + pVo8i k )v k . (11.9) 



f For example, we may be considering the motion of a body in a fluid through which a sound wave 
is propagated, the wavelength being large compared with the dimension of the body. 



36 Ideal Fluids 



§12 



We put the constant of integration equal to zero, since the velocity u of 
the body in its motion caused by the fluid must vanish when v vanishes. 
The relation obtained determines the velocity of the body from that of the 
fluid. If the density of the body is equal to that of the fluid (M = P V ), 
we have u = v, as we should expect. 



PROBLEMS 



sin°p UT i? N * ? omparin S f ("- 1 ) ^th the expression for * for flow past a sphere obtained in 
$1U, Problem 2, we see that 



where R is the radius of the sphere. The total momentum transmitted to the fluid bv the 
sphere is, according to (11.6), P = firpJPu, so that the tensor m t1c is 



• P™.™ 1 . Obtain i the equation of motion for a sphere executing an oscillatory motion 
in an ideal fluid, and for a sphere set in motion by an oscillating fluid. 

he expression i 

A = IRSvl, 

The total mon 
pF?u, so that 

Wik = frrpR 3 8nc. 
The drag on the moving sphere is 

F = -frrpl&du/dt, 
and the equation of motion of the sphere oscillating in the fluid is 

i^ipo+ip)^. = f, 

where Po is the density of the sphere. The coefficient of du/dtis the virtual mass of the sphere • 
it consists of the actual mass of the sphere and the induced mass, which in this case is half 
the mass of the fluid displaced by the sphere. 

If the sphere is set in motion by the fluid, we have for its velocity, from (1 1.9), 

3p 
u = v. 

p + 2po 

If the density of the sphere exceeds that of the fluid ( Po >/>),«< v, i.e. the sphere "lags 
behind the fluid; if Po < p, on the other hand, the sphere "goes ahead". 

Problem 2. Express the moment of the forces acting on a body moving in a fluid in 
terms of the vector A. 

Solution As we know from mechanics, the moment M of the forces acting on a body is 
iT™£ sa T lt8 e /f«ran«Wtt function (in this case, the energy E) by the relation 
7 . ' d ' where S0 ls the vector of an infinitesimal rotation of the body, and &E is the 
resulting change in E. Instead of rotating the body through an angle W (and correspondingly 
changing the components mic), we may rotate the fluid through an angle -80 relative to the 
body (and correspondingly change the velocity u). We have Su = — 80 xu, so that 

8E= P-Su = -SS-uxP. 
Using the expression (11.6) for P, we then obtain the required formula: 

M = -uxP = 4tt P A x u. 
§12. Gravity waves 

The free surface of a liquid in equilibrium in a gravitational field is a plane. 
If, under the action of some external perturbation, the surface is moved 



§12 Gravity waves 37 

from its equilibrium position at some point, motion will occur in the liquid. 
This motion will be propagated over the whole surface in the form of waves, 
which are called gravity waves, since they are due to the action of the gravita- 
tional field. Gravity waves appear mainly on the surface of the liquid, 
they affect the interior also, but less and less at greater and greater depths. 
We shall here consider gravity waves in which the velocity of the moving 
fluid particles is so small that we may neglect the term (v • grad)v in compari- 
son with dvjdt in Euler's equation. The physical significance of this is easily 
seen. During a time interval of the order of the period t of the oscillations 
of the fluid particles in the wave, these particles travel a distance of the order 
of the amplitude a of the wave. Their velocity is therefore of the order of 
afr. It varies noticeably over time intervals of the order of t and distances 
of the order of A in the direction of propagation (where A is the wavelength). 
Hence the time derivative of the velocity is of the order of v/t, and the space 
derivatives are of the order of vj\. Thus the condition (v • grad)v <^ dvjdt 
is equivalent to 



1 (a\ 2 a \ 

A \t/ t r 



or 

a < A, (12.1) 

i.e. the amplitude of the oscillations in the wave must be small compared with 
the wavelength. We have seen in §9 that, if the term (v«grad)v in the 
equation of motion may be neglected, we have potential flow. Assuming the 
fluid incompressible, we can therefore use equations (10.6) and (10.7). 
The term \v 2 in the latter equation may be neglected, since it contains the 
square of the velocity; putting f(t) = and including a term pgz on account 
of the gravitational field, we obtain 

P = -pgz-pd+Jdt. (12.2) 

We take the -sr-axis vertically upwards, as usual, and the ry-plane in the 
equilibrium surface of the fluid. 

Let us denote by £ the z co-ordinate of a point on the surface; £ is a func- 
tion of x, y and t. In equilibrium £ = 0, so that £ gives the vertical displace- 
ment of the surface in its oscillations. Let a constant pressure po (for example, 
the atmospheric pressure) act on the surface. Then we have at the surface, 
by (12.2), 

Po = —pgl — ptyjdt. 
Instead of the potential <j>, we can use a potential <j>' = </> + (pojp)t\ this makes 
no difference, since v = grad <f> = grad <f>'. The term p is removed from 
the above equation, however, and on dropping the prime we obtain the 
condition at the surface as 

gt+(d<f>jdt) z ^ = 0. (12.3) 

Since the amplitude of the wave oscillations is small, the displacement I 



38 Ideal Fluids §12 

is small. Hence we can suppose, to the same degree of approximation, that 
the vertical component of the velocity of points on the surface is simply the 
time derivative of £: 

But v z = 8<f>/8z, so that 

W/9*)z-t = dt/dt. 
Substituting £ from (12.3) we have 

\8z gdfl/^ 

Since the oscillations are small, we can take the value of the parenthesis 
at z = instead of z = £. Thus we have finally the following system of 
equations to determine the motion in a gravitational field: 

A<f> = 0, (12.4) 

(86 1 8U\ 

h^-dr =°- (12.5) 

\8z g 8t 2 7 Z=0 v } 

We shall here consider waves on the surface of a fluid whose area is un- 
limited, and we shall also suppose that the wavelength is small in comparison 
with the depth of the fluid; we can then regard the fluid as infinitely deep. 
We shall therefore omit the boundary conditions at the sides and bottom. 

Let us consider a gravity wave propagated along the #-axis and uniform 
in the ^-direction; in such a wave, all quantities are independent of y. We 
shall seek a solution which is a simple periodic function of time and of the 
co-ordinate x, i.e. we put 

<f> = f(z) cos (kx— cot). 

Here <o is what is called the circular frequency (we shall say simply the 
frequency) of the wave; l-njoa is the period of the motion at a given point; 
k is called the wave number; A = 2-njk is the wavelength, i.e. the period of the 
motion along the x-axis at a given time. 
Substituting in the equation 

8U 8% 
A<^ = —- + —- = 0, 

8x 2 8z 2 

we have 

d 2 //d#2-#2f = o. 

This equation has the solutions e ks and e~ kz . We must take the former, 
since the latter gives an unlimited increase of <f> as we go into the interior of 



§12 Gravity waves 39 

the fluid (we recall that the fluid occupies the region z < 0). Thus we obtain 
for the velocity potential 

<f> = Ae kz cos (kx- cot). (12.6) 

We have also to satisfy the boundary condition (12.5). Substituting (12.6), 
we obtain 

k-afi/g = 0, 
or 

ft>2 = kg. (12.7) 

This gives the relation between the wave number and the frequency of a 
gravity wave. 

The velocity distribution in the moving fluid is found by simply taking 
the space derivatives of <f> : 

v x = - Ake kz sin (kx - cot), v z = Ake kz cos (kx - cot). (12.8) 

We see that the velocity diminishes exponentially as we go into the fluid. 
At any given point in space (i.e. for given x, z) the velocity vector rotates 
uniformly in the ##-plane, its magnitude remaining constant and equal to 
Ake hz . 

Let us also determine the paths of fluid particles in the wave. We tem- 
porarily denote by x, z the co-ordinates of a moving fluid particle (and not 
of a point fixed in space), and by *o, %o the values of re and z at the equilibrium 
position of the particle. Then v x = d#/d*, v z = dz/dt, and on the right- 
hand side of (12.8) we may approximate by writing xo, zq in place of x, z, 
since the oscillations are small. An integration with respect to time then 
gives 

k 

x—xo= — A — e kz o cos (kxo — cot), 

CO 

k (12 - 9) 

z-zq = - A — e kz o sin (kx - cot). 

CO 

Thus the fluid particles describe circles of radius (Akjco)e h ^ about the points 
(*o, #o); this radius diminishes exponentially with increasing depth. 

The velocity of propagation U of the wave is, as we shall show in §66, 
U = Bwjdk. Substituting here <o = \/(kg), we find that the velocity of pro- 
pagation of gravity waves on an unbounded surface of infinitely deep fluid 
is 

u = W(g/k) = M£A/27r). v i2.i0) 

It increases with wavelength. 



40 Ideal Fluids §12 

PROBLEMS 

Problem 1. Determine the velocity of propagation of gravity waves on an unbounded 
surface of fluid of depth h. 

Solution. At the bottom of the fluid, the normal velocity component must be zero, 
i.e. v z = 8<f>ldz = f or z = —h. From this condition we find the ratio of the constants 
A and B in the general solution 

<f> = [Ae kz +Be-te]cos(kx-cot). 
The result is 

<f> = A cos(kx — cot) cosh. k(z+h). 

From the boundary condition (12.5) we find the relation between k and to to be 

co 2 — gk tanh kh. 
The velocity of propagation of the wave is 

1 / s r , ,, kh -\ 



— — tanhM + . 

V k tanh kh |_ cosh 2 kh J 

For M> 1 we have the result (12.10), and for kh < 1 the result (13.10) (see below). 



u = 

2 



Problem 2. Determine the relation between frequency and wavelength for gravity waves 
on the surface separating two fluids, the upper fluid being bounded above by a fixed horizontal 
plane, and the lower fluid being similarly bounded below. The density and depth of the 
lower fluid are p and h, those of the upper fluid are p' and h', and p > />'. 

Solution. We take the «y-plane as the equilibrium plane of separation of the two fluids. 
Let us seek a solution having in the two fluids the forms 

<f> = A cosh k{z+h) cos(kx— cot), 
4>' = B cosh. k(z—h')cos(kx— cot), 

so that the conditions at the upper and lower boundaries are satisfied; see the solution to 
Problem 1. At the surface of separation, the pressure must be continuous; by (12.2), this 
gives the condition 

8<f> 8cf)' 

Pgt+P— = Pgt+p'—- for z = £, 
ot ot 



i = ^( p 'i-^ t )- (2) 

Moreover, the velocity component v z must be the same for each fluid at the surface of separa- 
tion. This gives the condition 

8<f>/8z = dtfjdz for z = 0. (3) 

Now Vz = 8<f>/8z ~ dtjet and, substituting (2), we have 

86 8 2 <f>' 8 2 cf> 

5(P-P')- = P'^--^. (4) 

Substituting (1) in (3) and (4) gives two homogeneous linear equations for A and B, and the 



§12 Gravity waves 41 

condition of compatibility gives 

2 kg(p-p') 

CO* = 



p coth.kh-\-p coihhh' 

For kh^> 1, kh'^> 1 (both fluids very deep), 

p-p 



,2 = 



p + p 
while for kh <^ 1, kh' <^ 1 (long waves), 

g{p-p')hh' 



= k / g{p-py 

V ph' + p', 



ph' + p'h 

Problem 3. Determine the relation between frequency and wavelength for gravity waves 
propagated simultaneously on the surface of separation and on the upper surface of two fluid 
layers, the lower (of density p) being infinitely deep, and the upper (of density p') being of 
depth h' and having a free upper surface. 

Solution. We take the ary-plane as the equilibrium plane of separation of the two fluids. 
Let us seek a solution having in the two fluids the forms 

= Ae kz cos(&v- cot), 
(f)' = [Be~ kz +Ce kz ] cos(kx- cot). 
At the surface of separation, i.e. for z = 0, we have the conditions (see Problem 2) 

8$ af 8* ,s*f a^ 

- = ^. S{P-P}^ = P-^-^ (2) 

and at the upper surface, i.e. for z = h', the condition 

dz g dt z 

The first equation (2), on substitution in (1), gives A — C—B, and the remaining two con- 
ditions then give two equations for B and C; from the condition of compatibility we obtain a 
quadratic equation for o> 2 , whose roots are 

CO" 2 = &£ , CO* = kg. 

S p+ p ' + (p-p')e-2M> * 

For h' -*■ oo these roots correspond to waves propagated independently on the surface of 
separation and on the upper surface. 

Problem 4. Determine the possible frequencies of oscillationf (stationary waves) of a 
fluid of depth h in a rectangular tank of width a and length b. 

Solution. We take the * and y axes along two sides of the tank. Let us seek a solution 
in the form of a stationary wave: 

(f> = f(x,y) cosh k(z + h) cos cot. 

t See §68. 



42 Ideal Fluids §13 

We obtain for /the equation 

3 2 f 8 2 f 
dx 2 dy 2 
and the condition at the free surface gives, as in Problem 1, the relation 

co 2 = gk tanh kh. 
We take the solution of the equation for/ in the form 

/ = cos/w cos qy, p 2 + q 2 = k 2 . 
At the sides of the tank we must have the conditions 

v x = d<t>jdx = for x = 0, a; 
v y = 8<f>/dy = for y = 0, b. 

Hence we find p — mirfa, q = nn/b, where m, n are integers. The possible values of k % 
are therefore 



k 2 = 7T 2 l — + — 
U 2 b 2 J 



§13. Long gravity waves 

Having considered gravity waves of length small compared with the depth 
of the fluid, let us now discuss the opposite limiting case of waves of length 
large compared with the depth. These are called long waves. 

Let us examine first the propagation of long waves in a channel. The 
channel is supposed to be along the #-axis, and of infinite length. The 
cross-section of the channel may have any shape, and may vary along its 
length. We denote the cross-sectional area of the fluid in the channel by 
S = S(x, t). The depth and width of the channel are supposed small in 
comparison with the wavelength. 

We shall here consider longitudinal waves, in which the fluid moves along 
the channel. In such waves the velocity component v x along the channel is 
large compared with the components v y , v z . 

We denote v x by v simply, and omit small terms. The ^-component of 
Euler's equation can then be written in the form 

d<v 1 dp 

8t p dx 

and the ^-component in the form 

1 dp 

p dz 

we omit terms quadratic in the velocity, since the amplitude of the wave is 



§13 Long gravity waves 43 

again supposed small. From the second equation we have, since the pressure 
at the free surface (z = £) must be po, 

P =#)+#>(£ -4 
Substituting this expression in the first equation, we obtain 

dvjdt = -gdlfdx. (13.1) 

The second equation needed to determine the two unknowns v and £ can 
be derived similarly to the equation of continuity; it is essentially the equation 
of continuity for the case in question. Let us consider a volume of fluid 
bounded by two plane cross-sections of the channel at a distance dx apart. 
In unit time a volume (Sv) x of fluid flows through one plane, and a volume 
(Sv) x +ax through the other. Hence the volume of fluid between the two 
planes changes by 

8(Sv) 
(Sv) x+Ax -(Sv) x = — — dx. 

ox 

Since the fluid is incompressible, however, this change must be due simply to 
the change in the level of the fluid. The change per unit time in the volume 
of fluid between the two planes considered is (dSJdt)dx. We can therefore 
write 

8S 8(Sv) 

dx = dx, 

dt dx 
or 

dS 8(Sv) 

— + -— - = 0. (13.2) 

8t dx 

This is the required equation of continuity. 

Let .So be the equilibrium cross-sectional area of the fluid in the channel. 
Then S = So+S', where S' is the change in the cross-sectional area caused 
by the wave. Since the change in the fluid level is small, we can write S' 
in the form ££, where b is the width of the channel at the surface of the fluid. 
Equation (13.2) then becomes 

81 8(S v) 

b± + JlAjL = o. (13.3) 

8t 8x 

Differentiating (13.3) with respect to t and substituting 8v\8t from (13.1), 
we obtain 

8H g d I 8l\ _ 

~8&~bl)x\ °!hc) ~ ^ ' ' 

If the channel cross-section is the same at all points, then So = constant 
and 

8H gS e 2 £ 

-1 _ *_!? _± = o. (13.5) 

8& b 8x 2 v J 

This is called a wave equation: as we shall show in §63, it corresponds to 



44 Ideal Fluids §14 

the propagation of waves with a velocity U which is independent of frequency 
and is the square root of the coefficient of 8 2 £/8x 2 . Thus the velocity of propa- 
gation of long gravity waves in channels is 

U = V(gSo/b). (13.6) 

In an entirely similar manner, we can consider long waves in a large tank, 
which we suppose infinite in two directions (those of x and y). The depth 
of fluid in the tank is denoted by h. The component v z of the velocity is now 
small. Euler's equations take a form similar to (13.1): 

8v x dl dv y 81 

The equation of continuity is derived in the same way as (13.2) and is 

8h 8{hv x ) dihvy) 

8t dx 8y 

We write the depth h as h +£, where h is the equilibrium depth. Then 
$t d(hov x ) 8(hov y ) 

Let us assume that the tank has a horizontal bottom (ho = constant). 
Differentiating (13.8) with respect to t and substituting (13.7), we obtain 

TF- gh [l* + w)- ' (13 ' 9) 

This is again a (two-dimensional) wave equation; it corresponds to waves 
propagated with a velocity 

U = V(gfy- (13.10) 

§14. Waves in an incompressible fluid 

There is a kind of gravity wave which can be propagated inside an incom- 
pressible fluid. Such waves are due to an inhomogeneity of the fluid caused 
by the gravitational field. The pressure (and therefore the entropy s) neces- 
sarily varies with height; hence any displacement of a fluid particle in height 
destroys the mechanical equilibrium, and consequently causes an oscillatory 
motion. For, since the motion is adiabatic, the particle carries with it to its 
new position its old entropy s, which is not the same as the equilibrium value 
at the new position. 

We shall suppose below that the wavelength is small in comparison with 
distances over which the gravitational field causes a marked change in density; 
and we shall regard the fluid itself as incompressible. This means that we 
can neglect the change in its density caused by the pressure change in the 



§14 Waves in an incompressible fluid 45 

wave. The change in density caused by thermal expansion cannot be neglec- 
ted, since it is this that causes the phenomenon in question. 

Let us write down a system of hydrodynamic equations for this motion. 
We shall use a suffix to distinguish the values of quantities in mechanical 
equilibrium, and a prime to mark small deviations from those values. Then 
the equation of conservation of the entropy s = sq+s' can be written, to 
the first order of smallness, 

ds'ldt+vgradso = 0, (14.1) 

where so, like the equilibrium values of other quantities, is a given function 
of the vertical co-ordinate z. 

Next, in Euler's equation we again neglect the term (v • grad)v (since 
the oscillations are small); taking into account also the fact that the equili- 
brium pressure distribution is given by grad po = pog, we have to the same 
accuracy 

dv grad/) grad/>' grad/>o , 

St p po p 2 

Since, from what has been said above, the change in density is due only to 
the change in entropy, and not to the change in pressure, we can put 



\ &sq fp 



and we then obtain Euler's equation in the form 

*.«(*•) ,_**£. (14.2) 

dt po\ dso fp po 

We can take po under the gradient operator, since, as stated above, we always 
neglect the change in the equilibrium density over distances of the order of a 
wavelength. The density may likewise be supposed constant in the equation 
of continuity, which then becomes 

div v = 0. (14.3) 

We shall seek a solution of equations (14-1)— (14.3) in the form of a plane 
wave: 

v = constant x e Uk - r ~ ut \ 

and similarly for s' and />'. Substitution in the equation of continuity (14.3) 
gives 

vk = 0, (14.4) 

i.e. the fluid velocity is everywhere perpendicular to the wave vector k (a 
transverse wave). Equations (14.1) and (14.2) give 



teas — 



1 / 8 P0 \ , ik , 
= v«gradso> —iosw = — I I s g p . 

po \ «fro 1 v po 



46 Ideal Fluids §14 

The condition v • k = gives with the second of these equations 

** -(£)/■* 

and, eliminating v and s' from the two equations, we obtain the desired 
relation between the wave vector and the frequency, 

1 / dp \ ds 

Here and henceforward we omit the suffix zero to the equilibrium values of 
thermodynamic quantities; the #-axis is vertically upwards, and 6 is the 
angle between this axis and the direction of k. If the expression on the right 
of (14.5) is positive, the condition for the stability of the equilibrium distribu- 
tion s(z) (the condition that convection is absent — see §4) is fulfilled. 

We see that the frequency depends only on the direction of the wave 
vector, and not on its magnitude. For 6 = we have w = 0; this means 
that waves of the type considered, with the wave vector vertical, cannot 
exist. 

If the fluid is in both mechanical equilibrium and complete thermodynamic 
equilibrium, its temperature is constant and we can write 

^1 _ / 8s \ d P _ ( 6s \ 

dz~ \dp) T dz Pg \dpJ T ' 

Finally, using the well-known thermodynamic relations 

\8p) T p\dT) p * \ds) P CpKdTjp 
where c p is the specific heat per unit mass, we find 

/ T g I dp \ 

In particular, for a perfect gas, 



m ~vbo siaS - < i4 - 7 > 



CHAPTER II 

VISCOUS FLUIDS 

§15. The equations of motion of a viscous fluid 

Let us now study the effect of energy dissipation, occurring during the 
motion of a fluid, on that motion itself. This process is the result of the 
thermodynamic irreversibility of the motion. This irreversibility always 
occurs to some extent, and is due to internal friction (viscosity) and thermal 
conduction. 

In order to obtain the equations describing the motion of a viscous fluid, 
we have to include some additional terms in the equation of motion of an ideal 
fluid. The equation of continuity, as we see from its derivation, is equally 
valid for any fluid, whether viscous or not. Euler's equation, on the other 
hand, requires modification. 

We have seen in §7 that Euler's equation can be written in the form 

dt dxic 

where II <* is the momentum flux density tensor. The momentum flux 
given by formula (7.2) represents a completely reversible transfer of momen- 
tum, due simply to the mechanical transport of the different particles of fluid 
from place to place and to the pressure forces acting in the fluid. The viscosity 
(internal friction) is due to another, irreversible, transfer of momentum from 
points where the velocity is large to those where it is small. 

The equation of motion of a viscous fluid may therefore be obtained by 
adding to the "ideal" momentum flux (7.2) a term — o'ac which gives the 
irreversible "viscous" transfer of momentum in the fluid. Thus we write 
the momentum flux density tensor in a viscous fluid in the form 

Ilifc = pSik+pviVjc-a'ik = — Pffc+pOiZfe. (15.1) 

The tensor 

<*ik = —pbik+o'ik (15.2) 

is called the stress tensor, and o'tk the viscosity stress tensor, 0% gives the part 
of the momentum flux that is not due to the direct transfer of momentum 
with the mass of moving fluid.f 

The general form of the tensor a'ik can be established as follows. Processes 



f We shall see below that a' a contains a term proportional to Sa, i.e. of the same form as the 
term pSa. When the momentum flux tensor is put in such a form, therefore, we should specify what 
is meant by the pressure p; see the end of §49. 

47 



48 Viscous Fluids §15 

of internal friction occur in a fluid only when different fluid particles move 
with different velocities, so that there is a relative motion between various 
parts of the fluid. Hence a'ijc must depend on the space derivatives of the 
velocity. If the velocity gradients are small, we may suppose that the momen- 
tum transfer due to viscosity depends only on the first derivatives of the velo- 
city. To the same approximation, o'oc may be supposed a linear function of 
the derivatives dvt/dxk. There can be no terms in o'ac independent of 
dvijdxjc, since a'ik must vanish for v — constant. Next, we notice that o'tk 
must also vanish when the whole fluid is in uniform rotation, since it is clear 
that in such a motion no internal friction occurs in the fluid. In uniform rota- 
tion with angular velocity SI, the velocity v is equal to the vector product 
ftxr. The sums 

8vt dvjc 

dxjc Sxt 
are linear combinations of the derivatives dvi/dxic, and vanish when v = Slxr. 
Hence o'ik must contain just these symmetrical combinations of the deriva- 
tives dvi/dxjc. 

The most general tensor of rank two satisfying the above conditions is 

/ dvt 8v k \ 8v t 
a a = a -— + — - +b—-8 ik , 

\ OXjc OXi / oxi 

where a and b are independent of the velocity.f It is convenient, however, 
to write this expression in a slightly different form, in which a and b are 
replaced by other constants: 

o i* = t] — + — - |8«r— 1 +£8o—. (15.3) 

\ OXjc OXi OX\ I OXl 

The expression in parentheses has the property of vanishing on contraction 
with respect to i and k. The constants r\ and £ are called coefficients of viscosity. 
As we shall show in §§16 and 49, they are both positive: 

•n > 0, £ > 0. (15.4) 

The equations of motion of a viscous fluid can now be obtained by simply 
adding the expressions da'ujdxie to the right-hand side of Euler's equation 

/ 8vt dvi \ dp 

P\ 1" V]c = — 

*\ 8t dxjcf 



8xi 



Thus we have 



/ dvi dvi \ 

P h Vic 

\ 8t dxjcf 

dp 8 1/ 8vi 8vjc 8vi\\ 8 / dv t \ 

= - — + — U — + — -P**— ) + — U— • (15.5) 

8xi 8xjc \ \ dxjc 8xi 8xi J ) dxt \ 8xi J 



f In malang this statement we use the fact that the fluid is isotropic, as a result of which its proper- 
ties must be described by scalar quantities only (in this case, a and b). 



§15 The equations of motion of a viscous fluid 49 

This is the most general form of the equations of motion of a viscous fluid. 
The quantities rj and £ are functions of pressure and temperature. In 
general, p and T, and therefore rj and £, are not constant throughout the 
fluid, so that v\ and £ cannot be taken outside the gradient operator. 

In most cases, however, the viscosity coefficients do not change noticeably 
in the fluid, and they may be regarded as constant. We then have 

Sff'a; _ / &Vi 8 8v k 2 d dvi\ a dvi 

dxjc \ dxjcdxic dxi dxjc 3 dx\ dxi / dx% dxi 



8 2 Vi d dvi 

OXlJOXic OXi OXi 



But 



dvi/dxi = divv, d^i/dx^xjc == A»<- 
Hence we can write the equation of motion of a viscous fluid, in vector form, 

p\ f- (v«grad)v = -grad/> + i7Av + (£ + |^)graddivv. C15.6) 

If the fluid may be regarded as incompressible, div v = 0, and the last 
term on the right of (15.6) is zero. Thus the equation of motion of an 
incompressible viscous fluid is 

dv 1 7) 

h (v«grad)v = gradp + -A v. (15.7) 

dt p p 

This is called the Navier-Stokes equation. The stress tensor in an incom- 
pressible fluid takes the simple form 

(dv{ dvjc \ 
h— — I. (15.8) 
CXlc OXi I 

We see that the viscosity of an incompressible fluid is determined by only 
one coefficient. Since most fluids may be regarded as practically incompres- 
sible, it is this viscosity coefficient t] which is generally of importance. The 
ratio 

v = vIp (15.9) 

is called the kinematic viscosity (while rj itself is called the dynamic viscosity). 

We give below the values of ?) and v for various fluids, at a temperature of 

20° C: 

r\ (g/cm sec) v (cm 2 /sec) 

Water 0-010 0-010 

Air 0-00018 0-150 

Alcohol 0-018 0-022 

Glycerine 8-5 6-8 

Mercury 0-0156 0-0012 



50 Viscous Fluids §15 

It may be mentioned that the dynamic viscosity of a gas at a given tempera- 
ture is independent of the pressure. The kinematic viscosity, however, is 
inversely proportional to the pressure. 

The pressure can be eliminated from the Navier-Stokes equation in the 
same way as from Euler's equation. Taking the curl of both sides of equation 
(15.7), we obtain, instead of equation (2.11) as for an ideal fluid, 

8 
— (curl v) = curl (vx curl v) + vA(curlv). (15.10) 

We must also write down the boundary conditions on the equations of 
motion of a viscous fluid. There are always forces of molecular attraction 
between a viscous fluid and the surface of a solid body, and these forces have 
the result that the layer of fluid immediately adjacent to the surface is brought 
completely to rest, and "adheres" to the surface. Accordingly, the boundary 
conditions on the equations of motion of a viscous fluid require that the fluid 
velocity should vanish at fixed solid surfaces: 

v = 0. (15.11) 

It should be emphasised that both the normal and the tangential velocity 
component must vanish, whereas for an ideal fluid the boundary conditions 
require only the vanishing of v n .f 

In the general case of a moving surface, the velocity v must be equal to 
the velocity of the surface. 

It is easy to write down an expression for the force acting on a solid 
surface bounding the fluid. The force acting on an element of the surface is 
just the momentum flux through this element. The momentum flux through 
the surface element df is 

nW/& = (pvivjc— oi]c)6f}c. 

Writing &f k in the form d/& = tik df, where n is a unit vector along the normal, 
and recalling that v = at a solid surface, $ we find that the force P acting 
on unit surface area is 

Pi = -OiWUc = pni-a' ik n k . (15.12) 

The first term is the ordinary pressure of the fluid, while the second is the 
force of friction, due to the viscosity, acting on the surface. We must em- 
phasise that n in (15.12) is a unit vector along the outward normal to the fluid, 
i.e. along the inward normal to the solid surface. 

If we have a surface of separation between two immiscible fluids, the 
conditions at the surface are that the velocities of the fluids must be equal 

f We may note that, in general, Euler's equations cannot be satisfied with the boundary condi- 
tion v = 0. 

X In determining the force acting on the surface, each surface element must be considered in a 
frame of reference in which it is at rest. The force is equal to the momentum flux only when the 
surface is fixed. 



§15 The equations of motion of a viscous fluid 51 

and the forces which they exert on each other must be equal and opposite. 
The latter condition is written 

«l,fc oi,tt + «2,t 02, tk = 0, 

where the suffixes 1 and 2 refer to the two fluids. The normal vectors ni and 
112 are in opposite directions, i.e. »i f < = — »2,* = »*, so that we can write 

»< 01, ik = ni ai,ik- (15.13) 

At a free surface of the fluid the condition 

(Jikfik = a'iknic—pni = (15.14) 

must hold. 

We give below, for reference, expressions for the components of the stress 
tensor and the Navier-Stokes equation in cylindrical and spherical co- 
ordinates. In cylindrical co-ordinates r, <f>, z the components of the stress 
tensor are 

8v r /l 8v r 8v^ V 6 \ 

^=-^ + 2,—, ^ = ^__ + ___j, 

l\ dvt v r \ I 8v$ 1 8v z \ 

8v z I 8v z 8v r \ 

a. --,+2^, ,„.„(_ + _). (15 .15) 

The three components of the Navier-Stokes equation and the equation of 
continuity are 

8v r 8v r v 6 8v r 8v r v s 2 

+ v r + — — - + v z — 

8t 8r r 8(f> 8z r 

\8p t 8 2 v r 1 8 2 v r 8 2 v r 1 8v r 2 8v$ v r \ 

p 8r \ 8r 2 ~r* 8<f> 2 8z 2 r~8~r ~~ ~^~8~I~ "r 2 )' 



8V4 8v^ v* dv, to* v^ 

8t 8r r 8(f> 8z r 



1 dp /8 2 v 6 1 8 2 v# 8*vt 1 ty 2 dvr v+\ 
pr 8<f> \8r 2 r 2 8<f> 2 8z 2 r 8r r 2 8<f> r 2 V 



8v z 8v z Vx 8v z 8v z 

1- v r h — — - + v z 

8t 8r r 8$ 8z 



1 8p / 8 2 v z 1 8 2 v z 8 2 v z 1 8v z \ 
= \- v \ 1 1 1 I 

P 8z \ 8r 2 r 2 8<j> 2 8z 2 r 8r V 

8v r 1 dv+ 8v z v r ,---,* 

— + - + — + _ = o. (15.16) 

8r r 8j> 8z r J 



(15.17) 



52 Viscous Fluids §15 

In spherical co-ordinates r, <£, 9 we have for the stress tensor 

8v r 
8r 

°M = -P + 2 V ( — — -— + — + , 

\ r sin 8<p r r ] 

~ V \r 86 + ~8r~~7) y 

"" ^Usin^ 8<f> r 89 r F 

(dVj 1 d*, r w .\ 

\ dr r sm 9 8<f> r I 

while the equations of motion are 

8v r 8v r v 8v r v A 8v r v^+vJ 

1- v r 1 1 - L. 

8t dr r 86 rsin9 8<f> r 

\8p rl 

= + v \- 

p 8r lr 



Oaa = — ■ 



<JrO 



\8p pi 8 2 (rv r ) 1 8 2 v r 1 8 2 v r cot 6 8v r 



p 8r lr 8r 2 r 2 86* r 2 sin 2 9 8<f> 2 r 2 89 

2 8v e 



8vo_ 2 8v 4 2v r 2cot0 1 

89 ~ r 2 sin9^~~r^ r^~ T 



8v 8v e v d dv e v* 8v v r v e v A 2 cot6 
+ v r 1 1 ^ j. " * 



8t 8r r 89 rsin0 8(f> r 



— d JL J 1 8 ^ rv ^ 1 g % 1 8 2 v cot9 8v e 

pr 89 V [r 8r 2 r 2 89* r 2 sin 2 9 8<j> 2 + r 2 ~8~9 

2 cos 9 8v$ 2 8v r v e 



r 2 sin 2 9 8<f> r 2 89 r 2 sin 2 9 



]• 



8v 4> , dv <i> , *>0 &># V, 8v, VrV, V e V,COt9 

8t 8r r 89 r sin9 8<f> r r 

^__J g/> ri ^ 2 K) i a% l a% 

pr sin 9 8<f> V [r 8r 2 r 2 80 2 r 2 sin 2 9 8<j? + 

cot 9 8v 4> 2 8v r 2cos0 8v e v^ 



r 2 89 r 2 sin9 8<f> r 2 sin 2 9 8<f> r 2 sin 2 9 



]■ 



8v r 1 8v e 1 8v & 2v r v g cot9 

— + - ~- + —^~ a -~ + — + = 0. (15.18) 

8r r 89 rsm9 8j> r r J 



§16 Energy dissipation in an incompressible fluid 53 

Finally, we give the equation that must be satisfied by the stream function 
*fj(x,y) in two-dimensional flow of an incompressible viscous fluid. It is 
obtained by substituting v x = difj/dy, v y = -8tf,]dx, v z = in equation 
(15.10): 

8 (ai\ # g(A»A) , H g( A 0) 

^ Alfl) -Y x -^- + Vy-^r- v ^ = () - (15 ' 19) 

§16. Energy dissipation in an incompressible fluid 

The presence of viscosity results in the dissipation of energy, which is 
finally transformed into heat. The calculation of the energy dissipation is 
especially simple for an incompressible fluid. 

The total kinetic energy of an incompressible fluid is 

£kin = |pjVdF. 

We take the time derivative of this energy, writing 8(%pv 2 )ldt = pvtdvijdt 
and substituting for 8v\\8t the expression for it given by the Navier-Stokes 
equation : 

dvi _ dvi 1 dp 1 da'ik 



The result is 



8t dxjc p 8x% p dxjc 



d 1 da' 

— (w v2 ) = -pv«(vgrad)v-v«grad/>+^ 



dt ° dx k 

= - / ,(v.grad)(i^ + ^) + div(v.a')-cT^— . 
\ p I oxk 

Here v • a' denotes the vector whose components are Vio'uc- Since div v = 
for an incompressible fluid, we can write the first term on the right as a 
divergence : 

-(1^2) = _ div ^ v ^ 2 + t^j _ v . <J -o'i^. (16.1) 

The expression in brackets is just the energy flux density in the fluid: 
the term pv$v 2 +pjp) is the energy flux due to the actual transfer of fluid 
mass, and is the same as the energy flux in an ideal fluid (see (10.5)). The 
second term, v • o\ is the energy flux due to processes of internal friction. 
For the presence of viscosity results in a momentum flux a' a; a transfer 
of momentum, however, always involves a transfer of energy, and the 
energy flux is clearly equal to the scalar product of the momentum flux 
and the velocity. 



54 Viscous Fluids §16 



If we integrate (16.1) over some volume V, we obtain 

d 
Yt 



U P v*dV= - L Lv/i^ + ^-v.o'j.df- ( a' ik — dV. (16.2) 

The first term on the right gives the rate of change of the kinetic energy of 
the fluid in V owing to the energy flux through the surface bounding V. 
The integral in the second term is consequently the decrease per unit time 
in the kinetic energy owing to dissipation. 

If the integration is extended to the whole volume of the fluid, the surface 
integral vanishes (since the velocity vanishes at infinity-]-), and we find the 
energy dissipated per unit time in the whole fluid to be 

■Ekin = — cr'oc dV. 

J dxjc 

In incompressible fluids, the tensor cr'a is given by (15.8), so that 

dvi dvi 1 dvi dvjc \ 
a ik = 7] ( 1 J. 

dxjc 8x k \ dxjc 8xi 1 

It is easy to verify that this expression can be written 

/ 8vj dv k \ 2 

\ dx k dxt I 
Thus we have finally for the energy dissipation in an incompressible fluid 

• , C ( dvi dv k \ 2 

^--HU + id dF - (16 - 3) 

The < dissipation leads to a decrease in the mechanical energy, i.e. we must 
have E kia < 0. The integral in (16.3), however, is always positive. We 
therefore conclude that the viscosity coefficient rj is always positive. 

PROBLEM 

Transform the integral (16.3) for potential flow into an integral over the surface bounding 
the region of flow. 

Solution. Putting dv t /8xk — dvkldxt and integrating once by parts, we find 

^=- 2 "J(£) v =- 2 "J-£^ 



■E'kin = — t) J grada 2 «df. 



t We are considering the motion of the fluid in a system of co-ordinates such that the fluid is at 
rest at infinity. Here, and in similar cases, we speak, for the sake of definiteness, of an infinite volume 
of fluid, but this implies no loss of generality. For a fluid enclosed in a finite volume, the surface 
integral again vanishes, because the normal velocity component at the surface vanishes. 



§17 FUm in a pipe 55 

§17. Flow in a pipe 

We shall now consider some simple problems of motion of an incom- 
pressible viscous fluid. 

Let the fluid be enclosed between two parallel planes moving with a 
constant relative velocity u. We take one of these planes as the #;sr-plane, 
with the #-axis in the direction of u. It is clear that all quantities depend only 
on y, and that the fluid velocity is everywhere in the ^-direction. We have 
from (15.7) for steady flow 

dpjdy = 0, d 2 v/dy 2 = 0. 

(The equation of continuity is satisfied identically.) Hence p = constant, 
v = ay+b. For y — and y = h{h being the distance between the planes) 
we must have respectively v = and v = u. Thus 

v = yu/h. (17.1) 

The fluid velocity distribution is therefore linear. The mean fluid velocity, 
defined as 



1 h 
V = h J V dy ' 



is 



v = \u. (17.2) 

From (15.12) we find that the normal component of the force on either plane 
is just p, as it should be, while the tangential friction force on the plane 
y = is 

v xy = -q dv/dy = rju/h; (17.3) 

the force on the plane y = h is — rjujk. 

Next, let us consider steady flow between two fixed parallel planes in the 
presence of a pressure gradient. We choose the co-ordinates as before; 
the ff-axis is in the direction of motion of the fluid. The Navier-Stokes 
equations give, since the velocity clearly depends only on y, 

d 2 v 1 dp dp 

dy 2 r] 8x dy 

The second equation shows that the pressure is independent of y, i.e. it 
is constant across the depth of the fluid between the planes. The right-hand 
side of the first equation is therefore a function of x only, while the left-hand 
side is a function of y only; this can be true only if both sides are constant. 
Thus dpjdx — constant, i.e. the pressure is a linear function of the co-ordi- 
nate x along the direction of flow. For the velocity we now obtain 

1 dp 
v = — — y 2 + ay+b. 
2?7 dx 



56 Viscous Fluids §17 

The constants a and b are determined from the boundary conditions, v = 
for y = and jy = h. The result is 

°=-^Si A2 -0v-W]. (17.4) 

Thus the velocity varies parabolically across the fluid, reaching its maximum 
value in the middle. The mean fluid velocity (averaged over the depth of the 
fluid) is again 

1 h 



on calculating this, we find 

h* dp 

We may also calculate the frictional force a xy = r}(dv/dy)y= acting on one 
of the fixed planes. Substitution from (17.4) gives 

o xy = -\h dp/dx. (17.6) 

Finally, let us consider steady flow in a pipe of arbitrary cross-section 
(the same along the whole length of the pipe, however). We take the axis of 
the pipe as the ff-axis. The fluid velocity is evidently along the *-axis at all 
points, and is a function of y and z only. The equation of continuity is 
satisfied identically, while the y and z components of the Navier-Stokes 
equation again give dpjdy = dp/dz = 0, i.e. the pressure is constant over 
the cross-section of the pipe. The ^-component of equation (15.7) gives 

d"h) 8 z v 1 dp 
Qy2 Q z 2 yj dx 

Hence we again conclude that dp/dx = constant; the pressure gradient may 
therefore be written - Ap//, where Ap is the pressure difference between the 
ends of the pipe and / is its length. 

Thus the velocity distribution for flow in a pipe is determined by a two- 
dimensional equation of the form A v = constant. This equation has to be 
solved with the boundary condition v = at the circumference of the cross- 
section of the pipe. We shall solve the equation for a pipe of circular cross- 
section. Taking the origin at the centre of the circle and using polar co- 
ordinates, we have by symmetry v = v(r). Using the expression for the 
Laplacian in polar co-ordinates, we have 



1 d / dv\ Ap 

r dr \ dr J rjl 



§17 Flow in a pipe 57 

Integrating, we find 

Ap 
v= -r 2 + alogr + b. (17.8) 

The constant a must be put equal to zero, since the velocity must remain 
finite at the centre of the pipe. The constant b is determined from the re- 
quirement that v = for r = R, where R is the radius of the pipe. We then 
find 

v = -L{R2-r*). (17.9) 

Thus the velocity distribution across the pipe is parabolic. 

It is easy to determine the mass Q of fluid passing each second through 
any cross-section of the pipe (called the discharge). A mass p • 2tttv dV 
passes each second through an annular element litr dr of the cross-sectional 
area. Hence 

R 
Q = 2np I rvdr. 

Using (17.9), we obtain 

ttA/> 
Q = -£T&. (17.10) 

The mass of fluid is thus proportional to the fourth power of the radius of the 
pipe (Poiseuille's formula). 

PROBLEMS 

Problem 1. Determine the flow in a pipe of annular cross-section, the internal and external 
radii being R lt i? 8 . 

Solution. Determining the constants a and b in the general solution (17.8) from the con- 
ditions that v = for r — R t and r = R tt we find 



V = 



A M„, • . R*-Ri* 



r R 2 *-Ri* r 1 

lR 2 2_ r 2 + — t —log— . 

L log(i? 2 /i?i) g £ 2 J 



The discharge is 



■nAp [ (R 2 2 -Ri 2 ) 2 ! 

Q = — — #2 4 -#i 4 - — — 

^ 8v L loRiRtlRi) J 



log(Rt/Ri) 

Problem 2. The same as Problem 1, but for a pipe of elliptical cross-section. 

Solution. We seek a solution of equation (17.7) in the form v = Ay i +Bz a +C. The 
constants A, B, C axe determined from the requirement that this expression must satisfy 
the boundary condition » = 0on the circumference of the ellipse (i.e. Ay i +Bz i +C = 



58 Viscous Fluids §17 

must be the same as the equation y*la*+z*/b 2 = 1, where a and b are the semi-axes of the 
ellipse). The result is 

Ap a 2 b 2 

V = 



The discharge is 



Q = 



2r;l a* 

irAp a%^ 
4vl a* + b* 



Problem 3. The same as Problem 1, but for a pipe whose cross-section is an equilateral 
triangle of side a. 

Solution. The solution of equation (17.7) which vanishes on the bounding triangle is 

Ap 2 

where h u ha, h a are the lengths of the perpendiculars from a given point in the triangle to its 
three sides. For each of the expressions A/»i, AAj, M» (where A = 8 2 l8x a + 8 2 l8y 2 ) is 
zero; this is seen at once from the fact that each of the perpendiculars h lt h t , h a may be taken 
as the axis of y or z, and the result of applying the Laplacian to a co-ordinate is zero. We 
therefore have 

A(fah2h 3 ) = 2{hi grad V grad A 3 +/t 2 grad A 3 . grad Ai + 
+fa grad Ar grad A 2 ). 

But gradAj - n», gradA 2 = n 2 , gradAj = n S) where n^ n 2 , n s are unit vectors along 
the perpendiculars h lt h 2t h a . Any two of n lt n s , n 8 are at an angle 2tt/3, so that grad h x . grad hi 
= ni-nj = cos (2tt/3) = — £, and so on. We thus obtain the relation 

Aihfahs) = -{hi + h 2 + hz) = -JV3«, 

and we see that equation (17.7) is satisfied. The discharge is 

V3a 4 A/> 



= 



32(M 



Problem 4. A cylinder of radius i? x moves with velocity u inside a coaxial cylinder of 
radius R it their axes being parallel. Determine the motion of a fluid occupying the space 
between the cylinders. 

Solution. We take cylindrical co-ordinates, with the z-axis along the axis of the cylinders. 
The velocity is everywhere along the sr-axis and depends only on r (as does the pressure): 
v z — v(r). We obtain for v the equation 

1 d / dv\ 

r dr\ dr/ 

the term (vgrad)v = v 8v/8z vanishes identically. Using the boundary conditions v = u 
for r — JRj and v = for r — R 2 , we find 

log(r/U a ) 

v = u- 



logCRi/lfc) 

The frictional force per unit length of either cylinder is 277r^//log(i? 2 / J R 1 ). 



§17 Flow in a pipe 59 

Problem 5. A layer of fluid of thickness h is bounded above by a free surface and below 
by a fixed plane inclined at an angle a to the horizontal. Determine the flow due to gravity. 

Solution. We take the fixed plane as the *y-plane, with the *-axis in the direction of 
flow (Fig. 6). We seek a solution depending only on z. The Navier-Stokes equations with 
Vx — v{z) in a gravitational field are 

d 2 v dp 

r)— + pg sin* = 0, — + p£cosa = 0. 
dz 2 dz 

At the free surface (z = h) we must have a xt = ydv/dz = 0, a zt = — p = —po (.Po being 
the atmospheric pressure). For z = we must have v = 0. The solution satisfying these 
conditions is 

pg sin a 

p = po + pg(h - z) cos a, v = — z(2h - z). 

It] 

The discharge, per unit length in the y-direction, is 

pgh 3 sin a 

vdz = . 



= pJ 




Fig. 6 

Problem 6. Determine the way in which the pressure falls along a tube of circular cross- 
section in which a viscous perfect gas is flowing isothermally (bearing in mind that the 
dynamic viscosity t\ of a perfect gas is independent of the pressure). 

Solution. Over any short section of the pipe the gas may be supposed incompressible, 
provided that the pressure gradient is not too great, and we can therefore use formula 
(17.10), according to which 

<fr = SrjQ 

dx TTpR* 

Over greater distances, however, p varies, and the pressure is not a linear function of *. 
According to the equation of state, the gas density p = mplkT, where m is the mass of a 
molecule and k is Boltzmann's constant, so that 

dp _ SrjQkT 1 

dx irmR* p 

(The discharge Q of the gas through the tube is obviously the same, whether or not the gas 
is incompressible.) From this we find 

where p2, pi are the pressures at the ends of a section of the tube of length /. 



60 Viscous Fluids §18 

§18. Flow between rotating cylinders 

Let us now consider the motion of a fluid between two infinite coaxial 
cylinders of radii R lf R 2 (R 2 > i?i), rotating about their axis with angular 
velocities Q lf Q2. We take cylindrical co-ordinates r, <f>, z, with the s-axis 
along the axis of the cylinders. It is evident from symmetry that 

v z = v r = 0, V4 = v(r), p = p(r). 

The Navier-Stokes equation in cylindrical co-ordinates gives in this case two 
equations : 

dp/dr = pv 2 /r, (18.1) 

d 2 v 1 dv v 
dr 2 r dr r 2 

The latter equation has solutions of the form r n \ substitution gives n = ± 1, 
so that 

b 

v = ar + -. 
r 

The constants a and b are found from the boundary conditions, according to 
which the fluid velocity at the inner and outer cylindrical surfaces must be 
equal to that of the corresponding cylinder: v = Ri&i for r = i?i, v = R2O.2 
for r = R 2 . As a result we find the velocity distribution to be 

n 2 ^2 2 -^i^ 1 2 (ni-Q 2 )#iW 1 
" " Rf-Rf ' + W-RP ? (18 ' 3 > 

The pressure distribution is then found from (18.1) by straightforward 
integration. 

For Qi = &2 = & we have simply v = Dr, i.e. the fluid rotates rigidly 
with the cylinders. When the outer cylinder is absent (D.2 = 0, R 2 = 00) 
we have v = D.iRi 2 /r. 

Let us also determine the moment of the frictional forces acting on the 
cylinders. The frictional force acting on unit area of the inner cylinder is 
along the tangent to the surface and, from (15.12), is equal to the component 
a' r $ of the stress tensor. Using formulae (15.15), we find 

o (Q 1 -Q 2 )R 2 2 

— —2rt . 

' RJ-Rf 

The force acting on unit length of the cylinder is obtained by multiplying 



§19 The law of similarity 61 

by 2irRi, and the moment M\ of that force by multiplying the result by R\. 
We thus have 

WQi-Q 2 )fliW 

*-- *■-*. ■ (18 ' 4) 

The moment M2 of the forces acting on the inner cylinder is clearly — Mi.f 
The following general remark may be made concerning the solutions of the 
equations of motion of a viscous fluid which we have obtained in §§17 and 18. 
In all these cases the non-linear term (v • grad)v in the equations which 
determine the velocity distribution is identically zero, so that we are actually 
solving linear equations, a fact which very much simplifies the problem. 
For this reason all the solutions also satisfy the equations of motion for an 
incompressible ideal fluid, say in the form (10.2) and (10.3). This is why 
formulae (17.1) and (18.3) do not contain the viscosity coefficient at all. 
This coefficient appears only in formulae, such as (17.9), which relate the 
velocity to the pressure gradient in the fluid, since the presence of a pressure 
gradient is due to the viscosity; an ideal fluid could flow in a pipe even if 
there were no pressure gradient. 

§19. The law of similarity 

In studying the motion of viscous fluids we can obtain a number of impor- 
tant results from simple arguments concerning the dimensions of various 
physical quantities. Let us consider any particular type of motion, for 
instance the motion of a body of some definite shape through a fluid. If the 
body is not a sphere, its direction of motion must also be specified : e.g. the 
motion of an ellipsoid in the direction of its greatest or least axis. Alternatively, 
we may be considering flow in a region with boundaries of a definite form 
(a pipe of given cross-section, etc.). 

In such a case we say that bodies of the same shape are geometrically similar; 
they can be obtained from one another by changing all linear dimensions in 
the same ratio. Hence, if the shape of the body is given, it suffices to specify 
any one of its linear dimensions (the radius of a sphere or of a cylindrical 
pipe, one semi-axis of a spheroid of given eccentricity, and so on) in order 
to determine its dimensions completely. 

We shall at present consider steady flow. If, for example, we are discussing 
flow past a solid body (which case we shall take below, for definiteness), the 
velocity of the main stream must therefore be constant. We shall suppose 
the fluid incompressible. 

Of the parameters which characterise the fluid itself, only the kinematic 



f The solution of the more complex problem of the motion of a viscous fluid in a narrow space 
between cylinders whose axes are parallel but not coincident may be found in: N. E. Kochin, I. A. 
Kibel' and N. V. Roze, Theoretical Hydromechanics (Teoreticheskaya gidromekhanika), Part 2, 3rd 
ed., p. 419, Moscow 1948; A. Sommerfeld, Mechanics of Deformable Bodies, §36, Academic Press, 
New York 1950. 



62 ' Viscous Fluids §19 

viscosity v = rj/p appears in the equations of hydrodynamics (the Navier- 
Stokes equations); the unknown functions which have to be determined by 
solving the equations are the velocity v and the ratio pip of the pressure 
p to the constant density p. Moreover, the flow depends, through the 
boundary conditions, on the shape and dimensions of the body moving 
through the fluid and on its velocity. Since the shape of the body is supposed 
given, its geometrical properties are determined by one linear dimension, 
which we denote by /. Let the velocity of the main stream be u. Then any 
flow is specified by three parameters, v, u and /. These quantities have the 
following dimensions : 

v = cm 2 /sec, / = cm, u = cm/sec. 

It is easy to verify that only one dimensionless quantity can be formed from 
the above three, namely uljv. This combination is called the Reynolds 
number and is denoted by R: 

R = puljrj = uljv. (19.1) 

Any other dimensionless parameter can be written as a function of R. 

We shall now measure lengths in terms of /, and velocities in terms of u, 
i.e. we introduce the dimensionless quantities r//, v/w. Since the only 
dimensionless parameter is the Reynolds number, it is evident that the velocity 
distribution obtained by solving the equations of incompressible flow is 
given by a function of the form 

v = «f(r//, R). (19.2) 

It is seen from this expression that, in two different flows of the same type 
(for example, flow past spheres of different radii by fluids of different vis- 
cosities), the velocities vju are the same functions of the ratio x\l if the Reynolds 
number is the same for each flow. Flows which can be obtained from 
one another by simply changing the unit of measurement of co-ordinates and 
velocities are said to be similar. Thus flows of the same type with the same 
Reynolds number are similar. This is called the law of similarity (O. Rey- 
nolds 1883). 

A formula similar to (19.2) can be written for the pressure distribution in 
the fluid. To do so, we must construct from the parameters v y I, u some 
quantity with the dimensions of pressure divided by density; this quantity 
can be w 2 , for example. Then we can say that p/pu 2 is a function of the dimen- 
sionless variable r// and the dimensionless parameter R. Thus 

p = /o« 2 /(r//, R). (19.3) 

Finally, similar considerations can also be applied to quantities which 
characterise the flow but are not functions of the co-ordinates. Such a 
quantity is, for instance, the drag force F acting on the body. We can say 
that the dimensionless ratio of F to some quantity formed from v, «, /, p 



§20 Stokes' formula 63 

and having the dimensions of force must be a function of the Reynolds num- 
ber alone. Such a combination of v> u, I, p can be pu 2 l 2 , for example. Then 

F = pu*Pf(R). (19.4) 

If the force of gravity has an important effect on the flow, then the latter 
is determined not by three but by four parameters, /, u, v and the acceleration 
g due to gravity. From these parameters we can construct not one but two 
independent dimensionless quantities. These can be, for instance, the Rey- 
nolds number and the Froude number, which is 

F = u*llg. (19.5) 

In formulae (19.2)-(19.4) the function /will now depend on not one but two 
parameters (R and F), and two flows will be similar only if both these num- 
bers have the same values. 

Finally, we may say a little regarding non-steady flows. A non-steady flow 
of a given type is characterised not only by the quantities v, w, / but also 
by some time interval t characteristic of the flow, which determines the rate 
of change of the flow. For instance, in oscillations, according to a given law, 
of a solid body, of a given shape, immersed in a fluid, t may be the period of 
oscillation. From the four quantities v, u, /, r we can again construct two 
independent dimensionless quantities, which may be the Reynolds number 
and the number 

S = ut/1, (19.6) 

sometimes called the Strouhal number. Similar motion takes place in these 
cases only if both these numbers have the same values. 

If the oscillations of the fluid occur spontaneously (and not under the action 
of a given external exciting force), then for motion of a given type S will be 
a definite function of R: 

S=/(R). 



§20. Stokes' formula 

The Navier-Stokes equation is considerably simplified in the case of flow 
at small Reynolds numbers. For steady flow of an incompressible fluid, this 
equation is 

(v.grad)v= -(l//>)grad/> + (V/°)Av. 

The term (v • grad)v is of the order of magnitude of w 2 //, u and / having the 
same meaning as in §19. The quantity (rj/p) Av is of the order of magnitude 
of rju[pl 2 . The ratio of the two is just the Reynolds number. Hence the term 
(v • grad)v may be neglected if the Reynolds number is small, and the 
equation of motion reduces to a linear equation 

*?Av-grad/> = 0. (20.1) 



64 Viscous Fluids §20 

Together with the equation of continuity 

div v = (20.2) 

it completely determines the motion. It is useful to note also the equation 

A curlv = 0, (20.3) 

which is obtained by taking the curl of equation (20.1). 

As an example, let us consider rectilinear and uniform motion of a sphere 
in a viscous fluid. The problem of the motion of a sphere, it is clear, is 
exactly equivalent to that of flow past a fixed sphere, the fluid having a 
given velocity u at infinity. The velocity distribution in the first problem is 
obtained from that in the second problem by simply subtracting the velocity 
u; the fluid is then at rest at infinity, while the sphere moves with velocity 
-u. If we regard the flow as steady, we must, of course, speak of the flow 
past a fixed sphere, since, when the sphere moves, the velocity of the fluid at 
any point in space varies with time. 

Thus we must have v = u at infinity; we write v = v'-f u, so that v' 
is zero at infinity. Since div v = div v' = 0, v' can be written as the curl of 
some vector : v = curl A+ u. The curl of a polar vector is well known to be 
an axial vector, and vice versa. Since the velocity is an ordinary polar vector, 
A must be an axial vector. Now v, and therefore A, depend only on the radius 
vector r (we take the origin at the centre of the sphere) and on the parameter 
u; both these vectors are polar. Furthermore, A must evidently be a linear 
function of u. The only such axial vector which can be constructed for a 
completely symmetrical body (the sphere) from two polar vectors is the 
vector product rxu. Hence A must be of the form /'(r)nxu, where f(r) 
is a scalar function of r, and n is a unit vector in the direction of the radius 
vector. The product f(r)n can be written as the gradient, grad /(»-), of some 
function /(r), so that the general form of A is grad/xu. Hence we can write 
the velocity v' as 

v' = curl [grad/xu]. 

Since u is a constant, grad/xu = curl(/u), so that 

v = curl curl (/u) + u. (20.4) 

To determine the function /, we use equation (20.3). Since 
curlv = curl curl curl(/u) = (grad div- A)curl(/u) 
= -A curl(/u), 
(20.3) takes the form A 2 curl (Ju) = 0, or, since u = constant, 

A 2 (grad/xu) = (A 2 grad/)xu = 0. 
It follows from this that 

A 2 grad/ = 0. (20.5) 



§20 Stokes' formula 65 

A first integration gives 

A 2 / = constant. 

It is easy to see that the constant must be zero, since the velocity v must 
vanish at infinity, and so must its derivatives. The expression A 2 / contains 
fourth derivatives of /, whilst the velocity is given in terms of the second 
derivatives of/. Thus we have 



1 d 

A 2 /=- 

Hence 



1 d / d \ 
r* dr\ or J 



A/= 2a/r + A. 

The constant A must be zero if the velocity is to vanish at infinity. From 
A/ = 2a/r we obtain 

f=ar+b/r. (20.6) 

The additive constant is omitted, since it is immaterial (the velocity being 
given by derivatives of/). 

Substituting in (20.4), we have after a simple calculation 

u+n(u«n) 3n(u«n)-u 

v = M-a - - + b . (20.7) 

r r 3 

The constants a and b have to be determined from the boundary conditions : 
at the surface of the sphere (r = R), v = 0, i.e. 

u + n(u»n) 3n(u«n) — u 

\x-a + b — = 0. 

R R? 

Since this equation must hold for all n, the coefficients of u and n(u • n) 
must each vanish : 

a b a 3b 

— + — -1 = 0, + — = 0. 

R R? R R? 

Hence a = f i?, b = %R 3 . Thus we have finally 

f=lRr+lR?lr, (20.8) 

„ u + n(u«n) u-3n(u«n) 

v = - f # i L - IR* S L + u> (20.9) 

r r 3 

or, in spherical components, 

3R R3 



r 3R ft*! 

V r = U COS 1 1 , 

L 2r 2r3j' 

. A r 3R R3-] 

g — —u sin 1 . 

9 L 4r 4r3j 



(20.10) 



66 Viscous Fluids 



§20 



This gives the velocity distribution about the moving sphere. To determine 
the pressure, we substitute (20.4) in (20.1): 

gradp = -qAv = yj A curl curl (/u) 

= V A (grad div (/u) - u A/). 
But A 2 / =0, and so 

gradp = grad[7jAdiv(/u)] = gradfru-grad A/). 
Hence 

/> = rju . grad Af+po, (20. 1 1) 

where />o is the fluid pressure at infinity. Substitution for /leads to the final 
expression 

u*n 

P = Po - h-^-R. (20.12) 

Using the above formulae, we can calculate the force F exerted on the 
sphere by the moving fluid (or, what is the same thing, the drag on the sphere 
as it moves through the fluid). To do so, we take spherical co-ordinates with 
the polar axis parallel to u; by symmetry, all quantities are functions only of 
r and of the polar angle 0. The force F is evidently parallel to the velocity u. 
The magnitude of this force can be determined from (15.12). Taking from 
this formula the components, normal and tangential to the surface, of the 
force on an element of the surface of the sphere, and projecting these compo- 
nents on the direction of u, we find 

F = j>(-p cos 6+ a'rr cos - a rd sin 0)d/, (20. 13) 

where the integration is taken over the whole surface of the sphere. 
Substituting the expressions (20.10) in the formulae 

dr \r 86 dr r / 

(see (15.17)), we find that at the surface of the sphere 

o'rr = 0, a' re = - (3r)/2R)u sin 0, 

while the pressure (20.12) is p = p -(3 V l2R)u cos 0. Hence the integral 
(20.13) reduces to F = (3?/w/2#) § d/, or, finally,f 

F = 67TRr)u. (20.14) 

This formula (called Stokes' formula) gives the drag on a sphere moving 

f f T^on ^7 to u som e later applications, we may mention that, if the calculations are done with 
formula (20.7) for the velocity (the constants a and b being undetermined), we find 

F = 87rarju. (20.14a) 



§20 Stokes' formula 67 

slowly in a fluid. We may notice that the drag is proportional to the first 
powers of the velocity and linear dimension of the body.f 

This dependence of the drag on the velocity and dimension holds for 
slowly-moving bodies of other shapes also. The direction of the drag on a 
body of arbitrary shape is not the same as that of the velocity; the general 
form of the dependence of F on u can be written 

Ft = flan* (20.15) 

where aue is a tensor of rank two, independent of the velocity. It is important 
to note that this tensor is symmetrical {am — aid), a result which holds in the 
linear approximation with respect to the velocity, and is a particular case of 
a general law valid for slow motion accompanied by dissipative processes.! 
The solution that we have just obtained for flow past a sphere is not 
valid at great distances from it, even if the Reynolds number is small. In 
order to see this, we estimate the magnitude of the term (v • grad)v, which 
we neglected in (20.1). At great distances the velocity is u. The derivatives 
of the velocity at these distances are seen from (20.9) to be of the order of 
uR/r 2 . Thus (v • grad)v is of the order of u 2 R/r z . The terms retained in 
equation (20.1), for example (l//>) grad/>, are of the order r]Rujpr z (cf. (20.12)). 
The condition 

u-qR/prZ > uWlr 2 

holds only at distances r <^ vju, where v = rjjp. At greater distances, the 
terms we have omitted cannot legitimately be neglected, and the velocity 
distribution obtained is incorrect. 

To obtain the velocity distribution at great distances from the body, 
we have to take into account the term (v-grad)v omitted in (20.1). Since 
the velocity v is nearly equal to u at these distances, we can put approximately 
U'grad in place of v»grad. We then find for the velocity at great distances 
the linear equation 

(u-grad)v = -(l//o) grad/> + vAv (20.16) 

(C. W. Oseen, 1910). 
We shall not pause to give here the solution of this equation for flow 

f The drag can also be calculated for a slowly-moving ellipsoid of any shape. The corresponding 
formulae are given by H. Lamb, Hydrodynamics, 6th ed., §339, Cambridge 1932. We give here the 
limiting expressions for a plane circular disk of radius R moving perpendicular to its plane : 

F = \(yrjRu 

and for a similar disk moving in its plane: 

F = 32r)Ruj3. 

X See, for instance, Statistical Physics, §120, Pergamon Press, London 1958. 



6% Viscous Fluids 



§20 



past a sphere,f but merely mention that the velocity distribution thus 
obtained can be used to derive a more accurate formula for the drag on the 
sphere, which includes the next term in the expansion of the drag in powers 
of the Reynolds number uRJv. This formula is J 

F = (m^uRl 1 + -_ I. (20.17) 

Finally, we may mention that, in solving the problem of flow past an 
infinite cylinder with the main stream perpendicular to the axis of the 
cylinder, Oseen's equation has to be used from the start; in this case, equation 
(20.1) has no solution which satisfies the boundary conditions at the surface 
of the cylinder and at the same time vanishes at infinity. The drag per unit 
length of the cylinder is found to be 

4tttjU 

= h-y-log(uR/4 v y < 20 - 18 ) 

where y = 0-577 is Euler's constant. 



PROBLEMS 

Problem 1 Determine the motion of a fluid occupying the space between two concentric 
spheres of radii R lt R z (R 2 > RJ, rotating uniformly about different diameters with angular 
velocities R lt fi 2 ; the Reynolds numbers <W/", iW/" are small compared with unity. 

Solution On account of the linearity of the equations, the motion between two rotating 
spheres may be regarded as a superposition of the two motions obtained when one sphere is 
at rest and the other rotates. We first put fi a = 0, i.e. only the inner sphere is rotating. It 
is reasonable to suppose that the fluid velocity at every point is along the tangent to a circle 
m a plane perpendicular to the axis of rotation with its centre on the axis. On account of 
the axial symmetry, the pressure gradient in this direction is zero. Hence the equation of 
motion (20.1) becomes Av = 0. The angular velocity vector H, is an axial vector Argu- 
ments similar to those given previously show that the velocity can be written as 

v = curl[/(r)fti] = grad/x fli. 

The equation of motion then gives grad A/X Si x = 0. Since the vector grad A / is parallel 
to the radius vector, and the vector product rXfi, cannot be zero for given Si t and arbitrarv 
r, we must have grad A/ = 0, so that 

A/= constant. 



\ A I / ietai , Ied account of the calculations for a sphere and a cylinder is given by N E Kochin 
I. A. Kibel and N. V. Rozb, Theoretical Hydromechanics (Teoreticheskaya gidromekhanika), Part 2, 
brid C e 1932 aPtei " §§25 ~ 26, Moscow 1948 5 H - Lamb, Hydrodynamics, 6th ed., §§342-3, Cam- 

X At first sight it might appear that Osben's equation, which does not correctly give the velocity 
distribution near the sphere, could not be used to calculate the correction to the drag. In fact however 
the contribution to F due to the motion of the neighbouring fluid (where u < vlr) must be expanded in 
powers of the vector u. The first non-zero correction term in F arising from this contribution is 
then proportional to « 2 u, i.e. is of the second order with respect to the Reynolds number: it therefore 
does not affect the first-order correction in formula (20.17). Further corrections to Stokes' formula 
cannot be calculated from Oseen's formula. 



§20 Stokes' formula 69 

Integrating, we find 

f=arZ + -, v= ( — -2a|fiixr. 

The constants a and 6 are found from the conditions that v = for r = R 2 and v = u 
for r = R u where u = Si x xr is the velocity of points on the rotating sphere. The result is 

_ fli3J?2 3 / 1 1 \ 

#2 3 -#l 3 I 7» " 5^ j ^ X r " 

The fluid pressure is constant (p = p ). Similarly, we have for the case where the outer 
sphere rotates and the inner one is at rest (i^ = 0) 



V = 



R 2 s-R 

\P = Po)- 
i is at res 

R^R 2 3 



8/1 1 \ 

» 1 3\ j R 1 3 r 3) 



R 2 3 -Ri 

In the general case where both spheres rotate, we have 



V = 



RM 2 3 



'*[(*- h)* xt+ (-h-h)* Xi 



R 2 *-R 

If the outer sphere is absent (R 2 = 00, Q 2 = 0), i.e. we have simply a sphere of radius R 
rotating in an infinite fluid, then 

V = (#3/ r 3) Sl xr 

Let us calculate the moment of the frictional forces acting on the sphere in this case. If we 
take spherical co-ordinates with the polar axis parallel to SI, we have v r = v & = 0, v 6 — v 
= (R 3 £l/r 2 ) sin 0. The frictional force on unit area of the sphere is 



/ / 8v v\ 
a r 4> = 7][ =— 3r]Q, sin 6. 



The total moment on the sphere is 



M = j o' H Rsmd- 2ttR? sin 66, 




whence we find 



M= -SirrjRSQ. 



If the inner sphere is absent, v = £l 2 X r, i.e. the fluid simply rotates rigidly with the sphere 
surrounding it. 

Problem 2. Determine the velocity of a spherical drop of fluid (of viscosity •>?') moving 
under gravity in a fluid of viscosity 17 (W. Rybczynski 1911). 

Solution. We use a system of co-ordinates in which the drop is at rest. For the fluid 
outside the drop we again seek a solution of equation (20.5) in the form (20.6), so that the 
velocity has the form (20.7). For the fluid inside the drop, we have to find a solution which 
does not have a singularity at r = (and the second derivatives of/, which determine the 
velocity, must also remain finite). This solution is 



70 Viscous Fluids §20 

and the corresponding velocity is 

v = -^u4-£r 2 [n(u.n)-2u]. 

At the surface of the sphere t the following conditions must be satisfied. The normal velocity 
components outside (vg) and inside (v<) the drop must be zero: 

Vi,r = Ve t r = 0. 

The tangential velocity component must be continuous: 

Vl,d = Ve,6> 
as must be the component a T $ of the stress tensor : 

&i,r8 = a e,ro- 

The condition that the stress tensor components a rr are equal need not be written down; 
it would determine the required velocity u, which is more simply found in the manner shown 
below. From the above four conditions we obtain four equations for the constants a, b,A, B, 
whose solutions are 

a = R 2ri + 3ri ' . t = R > 1' .. . A--B*- V 



4(l+l')' 4{,+V)' 2(1+1') 

By (20.14a), we have for the drag 

F = 2TTWt]R{h] + 3r)')/(ri + r)'). 

As i\ -*■ oo Ccorresponding to a solid sphere) this formula becomes Stokes' formula. In the 
limit -q' -*■ (corresponding to a gas bubble) we have F = AtrwqR, i.e. the drag is two-thirds 
of that on a solid sphere. 

Equating F to the force of gravity on the drop, %nR z {p — p')g, we find 

2R ^{p-p')(r ) + rj') 

u = . 

3rj(2rj + 3r)') 

Problem 3. Two parallel plane circular disks (of radius R) lie one above the other a small 
distance apart; the space between them is filled with fluid. The disks approach at a constant 
velocity «, displacing the fluid. Determine the resistance to their motion (O. Reynolds). 

Solution. We take cylindrical co-ordinates, with the origin at the centre of the lower disk, 
which we suppose fixed. The flow is axisymmetric and, since the fluid layer is thin, pre- 
dominantly radial: v z <^.v r , and also 8v r /8r <^.8v T /dz. Hence the equations of motion 
become 

d*v r dp 8p 

1 dz* 8r dz ' 

lM + ^ = 0) (2) 

r dr dz 



f We may neglect the change of shape of the drop in its motion, since this change is of a higher 
order of smallness. However, it must be borne in mind that, in order that the moving drop should 
in fact be spherical, the forces due to surface tension at its boundary must exceed the forces due to 
pressure differences, which tend to make the drop non-spherical. This means that we must have 
•qujR <^ a.jR, where a is the surface-tension coefficient, or, substituting u ~ R^gpj-q, 

R < V(*!pg)- 



§21 The laminar wake 71 



with the boundary conditions 

at* = 

at z = h 
atr = R 



v r — v z = 0; 

v r = 0, v z = —u\ 

P = po, 



where h is the distance between the disks, and po the external pressure. From equations (1) 
we find 

1 dp 

Integrating equation (2) with respect to z, we obtain 

1 d r A3 a / dp \ 

u = I rv r az = \r — I, 

rdrj \Znrdr\&) 

o 

whence 

3inu 
P=Po + -^(R 2 -r*). 

The total resistance to the moving disk is 

F = 3irr)uR*/2hK 

§21. The laminar wake 

In steady flow of a viscous fluid past a solid body, the flow at great distances 
behind the body has certain characteristics which can be investigated inde- 
pendently of the particular shape of the body. 

Let us denote by U the constant velocity of the incident current; we take 
the direction of U as the *-axis, with the origin somewhere inside the body. 
The actual fluid velocity at any point may be written U+v; v vanishes at 
infinity. 

It is found that, at great distances behind the body, the velocity v is 
noticeably different from zero only in a relatively narrow region near the 
#-axis. This region, called the laminar zvake,f is reached by fluid particles 
which move along streamlines passing fairly close to the body. Hence the 
flow in the wake is essentially rotational. On the other hand, the viscosity has 
almost no effect at any point on streamlines that do not pass near the body, 
and the vorticity, which is zero in the incident current, remains practically 
zero on these streamlines, as it would in an ideal fluid. Thus the flow at 
great distances from the body may be regarded as potential flow everywhere 
except in the wake. 

We shall now derive formulae relating the properties of the flow in the 
wake to the forces acting on the body. The total momentum transported by 
the fluid through any closed surface surrounding the body is equal to the 



t In contradistinction to the turbulent wake; see §36. 



72 Viscous Fluids §21 

integral of the momentum flux density tensor over that surface, <j> H adfjc. 
The components of the tensor II $& are 

n« = p&ik+ p(Ui+Vi)(Uic+v k ). 

We write the pressure in the form p = po +p\ where po is the pressure at 
infinity. The integration of the constant term poSac+ pUtUjc gives zero, 
since the vector integral | df over a closed surface is zero. The integral 
Ui <j> pvjcdfk also vanishes : since the total mass of fluid in the volume con- 
sidered is constant, the total mass flux <j> pv«df through the surface surround- 
ing the volume must be zero. Finally, the velocity v far from the body is 
small compared with U. Hence, if the surface in question is sufficiently far 
from the body, we can neglect the term pvwk in 11^ as compared with 
pUjcVi. Thus the total momentum flux is 

<j> (p'Sik + pUjcViWk- 

Let us now take the fluid volume concerned to be the volume between two 
infinite planes x = constant, one of them far in front of the body and the 
other far behind it. The integral over the infinitely distant "lateral" surface 
vanishes (since p' — v = at infinity), and it is therefore sufficient to inte- 
grate only over the two planes. The momentum flux thus obtained is 
evidently the difference between the total momentum flux entering through 
the forward plane and that leaving through the backward plane. This 
difference, however, is just the quantity of momentum transmitted to the body 
by the fluid per unit time, i.e. the force F exerted on the body. 

Thus the components of the force F are 

^ = ( J7 - \\)(p'+ P Uv x )dydz, 

F ^ = ( J7 - j S )pUvydydz > 

F* = ( j f ~ j j )pUv z dydz, 

where the integration is taken over the infinite planes x = x± (far behind the 
body) and x = X2 (far in front of it). Let us first consider the expression for 
F x . 
Outside the wake we have potential flow, and therefore Bernoulli's equation 

p+%p(U+v) 2 = constant = p + %pU 2 
holds, or, neglecting the term \pv 2 in comparison with />U«v, 

p' = - P Uv x . 



JJ«*d>.d«-J'jgd V d,-0, Jjgd,d»-0, 



§21 The laminar wake 73 

We see that in this approximation the integrand in F x vanishes everywhere 
outside the wake. In other words, the integral over the plane x = #2 (which 
lies in front of the body and does not intersect the wake) is zero, and the 
integral over the plane x = xi need be taken only over the area covered by 
the cross-section of the wake. Inside the wake, however, the pressure change 
p' is of the order of pv 2 , i.e. small compared with pUv x . Thus we reach the 
result that the drag on the body is 

F x = - P UJjv x dydz, (21.1) 

where the integration is taken over the cross-sectional area of the wake far 
behind the body. The velocity v x in the wake is, of course, negative: the 
fluid moves more slowly than it would if the body were absent. Attention is 
called to the fact that the integral in (21.1) gives the amount by which the 
discharge through the wake falls short of its value in the absence of the body. 
Let us now consider the force (whose components are F y , F z ) which tends 
to move the body transversely. This force is called the lift. Outside the 
wake, where we have potential flow, we can write v y = 8<f>jdy, v z = dc/>J8z; 
the integral over the plane x = X2, which does not meet the wake, is zero: 

8< f> , /x f f d( f> 

dy 
since <j> = at infinity. We therefore find for the lift 

F y = -pUJj v y dy dz, F z = - P U jj v z dy dz. (21.2) 

The integration in these formulae is again taken only over the cross-sectional 
area of the wake. If the body has an axis of symmetry (not necessarily 
complete axial symmetry), and the flow is parallel to this axis, then the flow 
past the body has an axis of symmetry also. In this case the lift is, of course, 
zero. 

Let us return to the flow in the wake. An estimate of the magnitudes of 
various terms in the Navier-Stokes equation shows that the term »>A v can 
in general be neglected at distances r from the body such that rUjv > 1 
(cf . the derivation of the opposite condition at the beginning of §20) ; these 
are the distances at which the flow outside the wake may be regarded as 
potential flow. It is not possible to neglect that term inside the wake even 
at these distances, however, since the transverse derivatives 8 2 vfdy 2 , dfyjdz 2 
are large compared with dfy/dx 2 . 

The term (v«grad)v in the Navier-Stokes equation is of the order of mag- 
nitude (U+v)dvjdx ~ Uvjx in the wake. The term vAv is of the order 
of vd 2 v[dy 2 ~ w/Y 2 , where Y denotes the width of the wake, i.e. the order 
of magnitude of the distances from the #-axis at which the velocity v falls 
off markedly. If these two magnitudes are comparable, we find 

Y ~ ^(vxlU). (21.3) 



74 Viscous Fluids §21 

This quantity is in fact small compared with x, by the assumed condition 
Uxjv > 1. Thus the width of the laminar wake increases as the square root 
of the distance from the body. 

In order to determine how the velocity decreases with increasing x in the 
wake, we return to formula (21.1). The region of integration has an area of 
the order of Y 2 . Hence the integral can be estimated as F x ~ pUvY 2 , 
and by using the relation (21.3) we obtain 

v ~ Fzlpvx. (21.4) 

PROBLEMS 

Problem 1. Determine the flow in the laminar wake when there is both drag and lift. 

Solution. Writing the velocity in the Navier-Stokes equation in the form U+v and 
omitting terms quadratic in v (far from the body) we obtain 



— = -gra y-\+ v y— + —y y 



df 

we have also neglected the term 8 2 v/8x 2 in Av. We seek a solution in the form v = v 1 +v a> 
where v x satisfies 

dvi /d 2 vi 8 2 vi\ 
U — = v + . 

dx \ dy 2 dz 2 ) 

The term v 2 , which appears because of the term — grad(p/p) in the original equation, may be 
taken as the gradient grad O of some scalar. Since the derivatives with respect to x, far from 
the body, are small in comparison with those with respect to y and z, we may to the same 
approximation neglect the term 8Q>ldx in v x , i.e. take v x = v ix . 
Thus we have for v x the equation 



dv x / d*v x d*v x \ 
U = v - + r- . 

dx \ dy 2 dz 2 J 



This equation is formally the same as the two-dimensional equation of heat conduction, with 
x/U in place of the time, and the viscosity v in place of the thermometric conductivity. 
The solution which decreases with increasing y and z (for fixed *) and gives an infinitely 
narrow wake as * -»■ (in this approximation the dimensions of the body are regarded as 
small) is (see §51) 

F-r 1 

Vx= £_ _ e -l7<y , +* , >/4r* (1) 

Artpv X 

The constant coefficient in this formula is expressed in terms of the drag by means of 
formula (21.1), in which the integration over y and z may be extended to ±oo on account 
of the rapid decrease of v x . If we replace the Cartesian co-ordinates by spherical co-ordinates 
r, 9, <f> with the polar axis along the *-axis, then the region of the wake (V(y 2 + J8 *) <^*) 
corresponds to 9 <^ 1. In these co-ordinates formula (1) becomes 

ATrpv r 

The term dQ>/dx (with O given by formula (3) below), which we have omitted, would give 
a term in v x which diminishes more rapidly, as 1/r 2 . 



§21 The laminar wake 75 

v iv and v lz must have the same form as (1). We take the direction of the lift as the y-axis 
(so that F t = 0). According to (21.2) we have, since O — at infinity, 
oo oo 

J J Myd«-JJ («* + —)*>& 

—oo —oo 

= jj viy dydz = - FyjpU, 
jjvudydz = 0. 

Determining the constants in v iy and v\ t from these conditions, we find 

F v i , , ao 8® 

v y = -e- m^Vtox + 1 VgSS — . (2) 

Airpv x By Bz 

To determine the function O we proceed as follows. By the equation of continuity, 

div v « dvy/dy + dvzldz = 0; 

substituting (2), we have 

\ dy* dz 2 / By 

Differentiating this equation with respect to * and using the equation satisfied by v ly , we 
obtain 



/a 2 8 2 \8® a / dviy \ 

\ By 2 Bz 2 ) Bx ~ By\ Bx J 



---(■ 



Hence 



a 2 a 2 \ dviy 
+ — \ . 

ay 2 Bz 2 1 By 
BQ> v Bviy 



Bx U By 

Finally, substituting the expression for v iy and integrating with respect to x, we have 

F v V 
<I> = " Te-utf+zv&x- 1], (3) 

27TpU y 2 + z 2 

The constant of integration is chosen so that G> remains finite when y = z = 0. In spherical 
co-ordinates (with the azimuthal angle <f> measured from the xy-plane) 

F v cos 6 

It is seen from (2) and (3) that v y and v z , unlike v x , contain terms which decrease only as 1/0 2 
as we move away from the "axis" of the wake, as well as those which decrease exponen- 
tially with 6 (for a given r). 

The qualitative results (21.3) and (21.4) are, as we should expect, in agreement with the 
above formulae. If there is no lift, the flow in the wake is axially symmetrical. 



76 Viscous Fluids §22 

Problem 2. Determine the flow outside the wake far from the body. 

Solution. Outside the wake we assume potential flow. Since we are interested only in 
the terms in the potential <E> which decrease least rapidly with distance, we seek a solution of 
Laplace's equation A$ = as a sum of two terms: 

a cos<£ 

r r 

of which the first is centrally symmetric and belongs to the force F x , while the second is 
symmetrical about the xy-plane and belongs to the force F v . 

Using the expression for A® in spherical co-ordinates, we obtain for the function 
/(#) the equation 



— (sin0— ) 

dd\ del 



J » _£_ = o. 
sin 6 



The solution of this equation finite as 6 -> n is/ = b cot £0. The coefficient b must be deter- 
mined so as to give the correct value of F y . It is simpler, however, to use the fact that in the 
range V( v /Ur) <^ 8 <^ 1 this part of <S> must be the same as the expression 

o = Fy C0S ^ 

TmpU r6 ' 

obtained from formula (3'), Problem 1, for O in the wake. Hence b = FJAnpU. 

To determine the coefficient a, we notice that the total mass flux through a sphere S of 
large radius r equals zero, as for any closed surface. The rate of inflow through the part S 
of 5 intercepted by the wake is 

- JJvxdydz = F x jpU. 

S 

Hence the same quantity must flow out through the rest of the surface of the sphere, i.e. we 
must have 

<f vdf= F X / P U. 
s-s 

Since S is small compared with S, we can put 

j>vd{ = ^gradO-df = -Ana = F x j P U, 
s s 

whence a = —FxftnpU. 
The complete solution is given by the sum of these two expressions : 

1 

which gives the flow everywhere outside the wake far from the body. The potential dimini- 
shes with increasing distance as 1/r; the velocity v, therefore, diminishes as 1/r 2 . If there is 
no lift, the flow outside the wake is spherically symmetrical. 

§22. The viscosity of suspensions 

A fluid in which numerous fine solid particles are suspended (forming a 



§22 The viscosity of suspensions 77 

suspension) may be regarded as an homogeneous medium if we are concerned 
with phenomena whose characteristic lengths are large compared with the 
dimensions of the particles. Such a medium has an effective viscosity r\ 
which is different from the viscosity rjo of the original fluid. The value of rj 
can be calculated for the case where the concentration of the suspended 
particles is small (i.e. their total volume is small in comparison with that 
of the fluid). The calculations are relatively simple for the case of spherical 
particles (A. Einstein, 1906). 

It is necessary to consider first the effect of a single solid globule, immersed 
in a fluid, on flow having a constant velocity gradient. Let the unperturbed 
flow be described by a linear velocity distribution 

vot = a-ikX/c, (22.1) 

where a^ is a constant symmetrical tensor. The fluid pressure is constant: 

Po = constant, 

and in future we shall take po to be zero, i.e. measure only the deviation 
from this constant value. If the fluid is incompressible (div vo = 0), the 
sum of the diagonal elements of the tensor a^ must be zero : 

am = 0. (22.2) 

Now let a small sphere of radius R be placed at the origin. We denote 
the altered fluid velocity by v = vo + Vi ; vi must vanish at infinity, but near 
the sphere vi is not small compared with Vo. It is clear from the symmetry 
of the flow that the sphere remains at rest, so that the boundary condition is 
v = for r = R. 

The required solution of the equations of motion (20.1) to (20.3) may be 
obtained at once from the solution (20.4), with the function /given by (20.6), 
if we notice that the space derivatives of this solution are themselves solutions. 
In the present case we desire a solution depending on the components of the 
tensor a^ as parameters (and not on the vector u as in §20). Such a solution 
is 

vi = curl curl [(ocgrad)/], p = rjoCLwd 2 Afjdxidxn, 

where (a »grad)/ denotes a vector whose components are oLnfifjdxjc. Expand- 
ing these expressions and determining the constants a and b in the function 
/ = ar + bjr so as to satisfy the boundary conditions at the surface of the 
sphere, we obtain the following formulae for the velocity and pressure : 

5/RS R*\ #5 

vu = - — WkVUnm -ocm;%, (22.3) 

2 \ r 4 r l l r 4 

p = - 5-170— - (x-ikninjc, (22 A) 

r 3 

where n is a unit vector in the direction of the radius vector. 



78 Viscous Fluids §22 

Returning now to the problem of determining the effective viscosity of a 
suspension, we calculate the mean value (over the volume) of the momentum 
flux density tensor 11^, which, in the linear approximation with respect 
to the velocity, is the same as the stress tensor — a tt : 

*» = (1/V) j <j ik dV. 

The integration here may be taken over the volume V of a sphere of large 
radius, which is then extended to infinity. 
First of all, we have the identity 



/ dv t dv k \ 

°ik = yo — + — - I -pS ik + 

\ OXjc OXi I 

f /(•*-*(£ + 5) ^K (22 - 5) 



+ 



The integrand on the right is zero except within the solid spheres; since 
the concentration of the suspension is supposed small, the integral may be 
calculated for a single sphere as if the others were absent, and then multiplied 
by the concentration c of the suspension (the number of spheres per unit 
volume). The direct calculation of this integral would require an investi- 
gation of internal stresses in the spheres. We can circumvent this difficulty, 
however, by transforming the volume integral into a surface integral over an 
infinitely distant sphere, which lies entirely in the fluid. To do so, we note 
that the equation of motion daujdxi = leads to the identity 

<*ik = 8(cruXk)ldxi; 
hence the transformation of the volume integral into a surface integral gives 



<*ik 



I dvi dv k \ r 
= ^o(— + — l+cd) {auXkdfi-Tjo^idfk + vjcdfi)}. 



We have omitted the term in p, since the mean pressure is necessarily zero ; 
p is a scalar, which must be given by a linear combination of the components 
onijc, and the only such scalar is ecu = 0. 

In calculating the integral over a sphere of very large radius, only the 
terms of order 1/r 2 need be retained in the expression (22.3) for the velocity. 
A simple calculation gives the value of the integral as 



cr]o • 207rR s {5a.i m ninjcnin m —aan]cni}, 
where the bar denotes an average with respect to directions of the unit vector 



§23 Exact solutions of the equations of motion for a viscous fluid 79 

n. Effecting the averaging, j- we finally have 

(dVi dvjc \ 
1 I + 577o«» '%ttR z c. (22.6) 
oxjc 8xi J 

The ratio of the second term to the first determines the required relative 
correction to give the effective viscosity of the suspension. If we are in- 
terested only in corrections of the first order of smallness, we can take the 
first term as 2^oa«Ar- We then obtain for the effective viscosity of the suspen- 
sion 

r) = r}0(l+m> (22.7) 

where <f> = &nR z c is the small ratio of the total volume of the spheres to 
the total volume of the suspension. 

§23. Exact solutions of the equations of motion for a viscous fluid 

If the non-linear terms in the equations of motion of a viscous fluid do 
not vanish identically, the solving of these equations offers great difficulties, 
and exact solutions can be obtained only in a very small number of cases. 
Furthermore, it has not yet proved possible to carry out a complete investi- 
gation of the steady flow of a viscous fluid in all space round a body in the 
limit of very large Reynolds numbers. Although, as we shall see, such 
a flow does not in practice remain steady, the solution of the problem would 
nevertheless be of great methodological interest. $ 

We give below examples of exact solutions of the equations of motion for 
a viscous fluid. 

(1) An infinite plane disk immersed in a viscous fluid rotates uniformly 
about its axis. Determine the motion of the fluid caused by this motion of 
the disk (T. von KArman, 1921). 

We take cylindrical co-ordinates, with the plane of the disk as the plane 
z = 0. Let the disk rotate about the #-axis with angular velocity Q. We 
consider the unbounded volume of fluid on the side z > 0. The boundary 
conditions are 



*V = 0, 


»* = Qr, 


v z = for z = 0, 


v r = 0, 


^ = 


for z = oo. 



t The required mean values of products of components of the unit vector are symmetrical tensors, 
which can be formed only from the unit tensor S,*. We then easily find 

ntfljc = hoc, 



ninicmn m = Tt(8ik$lm + 8il$km + $im$ki)- 

X The "vanishing viscosity" theory of Oseen is concerned with this problem; it is unsatis- 
factory, since it is based on an unjustified simplification of the Navier-Stokes equations. Prandtl's 
boundary-layer theory (see §39) does not solve the problem throughout the volume of the fluid. 



80 



Viscous Fluids 



§23 



The axial velocity v z does not vanish as z -> oo, but tends to a constant nega- 
tive value determined by the equations of motion. The reason is that, 
since the fluid moves radially away from the axis of rotation, especially near 
the disk, there must be a constant vertical flow from infinity in order to 
satisfy the equation of continuity. We seek a solution of the equations of 
motion in the form 



(23.1) 



v r = rQF(z!); ^ = rQG(*i); v z = i/(yQ)H(xi); 

p — —pv£lP(zi) y where z\ = -\/(Q/v)z. 

In this velocity distribution, the radial and azimuthal velocities are propor- 
tional to the distance from the axis of rotation, while v z is constant on each 
horizontal plane. 




Substituting in the Navier-Stokes equation and in the equation of con- 
tinuity, we obtain the following equations for the functions F, G, H and P: 

F2_ G*+F'H = F", 2FG+ G'H = G", 

(23 2^ 
HH' = P'+H", 2F+H' = 0; V ' ' 

the prime denotes differentiation with respect to z\. The boundary conditions 
are 

F = 0, G = 1, H = for *i = 0. 

(23.3) 

F = 0, G = for ^i = oo. v } 

We have therefore reduced the solution of the problem to the integration of a 
system of ordinary differential equations in one variable ; this can be achieved 
numerically.f Fig. 7 shows the functions F, G and — H thus obtained. 



t The numerical integration has also been carried out for another similar problem, in which the 
fluid rotates uniformly at infinity and the disc is at rest (U.T. Bodewadt, Zeitschrift fur angeviandte 
Mathematik und Mechanik 20, 241, 1940). 



§23 Exact solutions of the equations of motion for a viscous fluid 81 

The limiting value of H as zi -> oo is -0-886; in other words, the fluid 
velocity at infinity is fl z (oo) = — Q-886\/(vQ). 

The frictional force acting on unit area of the disk perpendicularly to the 
radius is a z<f) = rj{dv^dz) z= Q. Neglecting edge effects, we may write the 
moment of the frictional forces acting on a disk of large but finite radius R as 

R 
M = 2 j 2rrr 2 a H Ar = 7r#W0^ 3 )G'( )- 
o 
The factor 2 in front of the integral appears because the disk has two sides 
exposed to the fluid. A numerical calculation of the function G leads to 
the formula 

M = - 1-94 JRW("Q 8 )- ( 23 - 4 ) 

(2) Determine the steady flow between two plane walls meeting at an 
angle a (Fig. 8 shows a cross-section of the two planes); the fluid flows 
out from the line of intersection of the planes (G. Hamel, 1916). 




Fig. 8 



We take cylindrical co-ordinates r, z, <f>, with the z-axis along the line of 
the intersection of the planes (the point O in Fig. 8), and the angle <£ measured 
as shown in Fig. 8. The flow is uniform in the ^-direction, and we naturally 
assume it to be entirely radial, i.e. 

i>4 = Vz = 0, v r = v(r, <f>). 

The equations (15.16) give 

dv 1 dp I d 2 v 1 d 2 v 1 dv v \ 

*,_.= L + J — + + -, (23.5) 

8r pdr \ dr 2 r d<f> r dr r 2 / 

1 dp 2v dv ,_ , x 

--£ + ^^r ' (23 - 6) 

pr d<f> r* d<j> 
d(rv)jdr = 0. 

It is seen from the last of these that rv is a function of <f> only. Introducing 
the function 

u(<f>) = rvl6v, (23.7) 



82 


Viscous Fluids 


we obtain from (23.6) 






1 dp \2v* du 
pd<j>~ r2 d<j>' 


whence 






p 12l/ 2 



§23 



ft 

Substituting this expression in (23.5), we have 

dhi 1 

_ + 4 „ + 6«s = —,*/<(,), 

from which we see that, since the left-hand side depends only on <£ and the 
right-hand side only on r, each must be a constant, which we denote by 2C\. 
Thus /'(*■) = 12v 2 Ci/r 3 , whence /(r) = -6v 2 Ci/r 2 + constant, and we have 
for the pressure 

p 6v 2 

- = —r-(2u - d) + constant. (23 .8) 

p r z 

For u(<f>) we have the equation 

m" + 4m + 6« 2 = 2Ci, 

which, on multiplication by u' and one integration, gives 

|m' 2 + 2 m 2 + 2 w 3_2Cim-2C 2 = 0. 
Hence we have 

24 = ± + C 3 , (23.9) 

which gives the required dependence of the velocity on <f>; the function u(<f>) 
can be expressed in terms of elliptic functions. The three constants Ci, C2, 
Cz are determined from the boundary conditions 

«(±i<x) = (23.10) 

and from the condition that the same mass Q of fluid passes in unit time 
through any cross-section r = constant: 



a/2 a/2 

J vrd<f> = 6vp J 

-a/2 -a/2 



Q = p j vrdt = 6v P j ud<f>. (23.11) 



Q may be either positive or negative. If Q > 0, the line of intersection 
of the planes is a source, i.e. the fluid emerges from the vertex of the angle : 
this is called flow in a diverging channel. If Q < 0, the line of intersection is 



§23 Exact solutions of the equations of motion for a viscous fluid 83 

a sink, and we have flow in a converging channel. The ratio \Q\jvp is dimen- 
sionless and plays the part of the Reynolds number in the problem considered. 
Let us first discuss converging flow (Q < 0). To investigate the solution 
(23.9)-(23.11) we make the assumptions, which will be justified later, that 
the flow is symmetrical about the plane <f> = (i.e. u(<f>) = u( — <f>)), and that 
the function u(4>) is everywhere negative (i.e. the velocity is everywhere 
towards the vertex) and decreases monotonically from u = at <j> = ± fa 
to u = — «o < at 4> = 0, so that uq is the maximum value of \u\. Then 
for u = —wo we must have dufd</> = 0, whence it follows that u = — #o 
is a zero of the cubic expression under the radical in the integrand of (23.9). 
We can therefore write 

— u z —u 2 +Ciu+C2 = (u + uo){ — u 2 -(l-uo)u + q} f 

where q is another constant. Thus 

u 

C dw 

26 = + -, (23.12) 

^ "J V\(u + u ){-u*-a-uo)u + q\y 



V[( M + «o){ - m 2 - (1 - mo)w + q}] ' 

the constants uq and q being determined from the conditions 

o 
a= f 



-«, 



du 



J VKu 



V[(« + «o){ - m 2 - (1 - u )u + q}] ' 

(23.13) 
udu 



\/[(u + u ){ - m 2 - (1 - u )u + q}] 

1*0 



Fig. 9 



(R = \Q\jvp)\ the constant q must be positive, since otherwise these integrals 
would be complex. The two equations just given may be shown to have 
solutions uq and q for any R and a < it. In other words, convergent sym- 
metrical flow (Fig. 9) is possible for any aperture angle a and any Reynolds 



84 Viscous Fluids §23 

number. Let us consider in more detail the flow for very large R. This 
corresponds to large uq. Writing (23.12) (for <j> > 0) as 

j 

r d« 

2(|a-<£) = , 

J \Z[(u + uo){-u 2 -(l-u )u + q}] 

we see that the integrand is small throughout the range of integration if \u \ 
is not close to uq. This means that \u | can differ appreciably from uq only 
for <f> close to ±^a, i.e. in the immediate neighbourhood of the walls.f 
In other words, we have u « constant = — uq for almost all angles <f>, and 
in addition uq = R/6a, as we see from equations (23.13). The velocity v 
itself is \Q |/potr, giving a non-viscous potential flow with velocity independent 
of angle and inversely proportional to r. Thus, for large Reynolds numbers, 
the flow in a converging channel differs very little from potential flow of 
an ideal fluid. The effect of the viscosity appears only in a very narrow layer 
near the walls, where the velocity falls rapidly to zero from the value cor- 
responding to the potential flow (Fig. 10). 




Fig. 10 



Now let Q > 0, so that we have divergent flow. At first we again suppose 
that the flow is symmetrical about the plane <j> = 0, and that u{<j>) (where 
now u > 0) varies monotonically from zero at cf> = ± |a to uq > at <j> = 0. 
Instead of (23.13) we now have 



■-/ 



«0 , 

QU 



V[(«o - u){u 2 + (1 + uq)u + q}] ' 

(23.14) 
r udu 

J vT( M o — u){u 2 + (l + uo)u + q}] 



f The question may be asked how the integral can cease to be small, even if u 7H — u . The answer 
is that, for u very large, one of the roots of — w 2 —(1 — w )u+g = is close to — u , so that the radicand 
has two almost coincident zeros, the whole integral therefore being "almost divergent" at u = —u . 



§23 Exact solutions of the equations of motion for a viscous fluid 85 

If we regard u as given, then a increases monotonically as q decreases, and 
takes its greatest value for q = 0: 

du 

CCmax 



r du 

J a/\u(uo-u)(u- 



\/[u(uq - u)(u + UQ + 1)] 






Fig. 11 



It is easy to see that for given q, on the other hand, a is a monotonically 
decreasing function of uq. Hence it follows that uq is a monotonically de- 
creasing function of q for given a, so that its greatest value is for q = 
and is given by the above equation. The maximum R = Rmax corresponds 
to the maximum wo- Using the substitutions k 2 = uol(l + 2uo), u = uq cos 2 x, 
we can write the dependence of R ma x on a in the parametric form 

ax 



C Q-X 

a = 2V(l-2tf) — — — — , 

J y\l— k z sm*x) 



1-&2 12 

Rmax = — 6a- — — + 





tt/2 



(23.15) 



1-2*2 V(l-2£ 2 ) 



\/(l — k 2 sin 2 x)dx. 



Thus symmetrical flow, everywhere divergent (Fig. 11a), is possible for a 
given aperture angle only for Reynolds numbers not exceeding a definite 
value. As a -> tr (k -> 0), Rmax -> 0; as a -> (k -> l/\/2)> Rmax tends to 
infinity as 18-8/a. 

For R > Rmax the assumption of symmetrical flow, everywhere divergent, 
is unjustified, since the conditions (23.14) cannot be satisfied. In the range 
of angles — |a < <f> ^ \on the function u(<f>) must now have maxima or 
minima. The values of u(<f>) corresponding to these extrema must again be 
zeros of the polynomial under the radical sign. It is therefore clear that 
the trinomial u 2 + (l+Uo)u + q (with uq > 0, q > 0) must have two real 
negative roots in the range mentioned, so that the radicand can be written 
{uq — u)(u+uo')(u+uo"), where uq > 0, uq' > 0, uq" > 0; we suppose 



86 Viscous Fluids §23 

wo' < «o". The function u{<j>) can evidently vary in the range mo > u ^ — uq\ 
u = uo corresponding to a positive maximum of u(<f>), and u — -«o' to a 
negative minimum. Without pausing to make a detailed investigation of the 
solutions obtained in this way, we may mention that for R > R m ax a solution 
appears in which the velocity has one maximum and one minimum, the flow 
being asymmetric about the plane <f> = (Fig. lib). When R increases fur- 
ther, a symmetrical solution with one minimum and two maxima appears 
(Fig. lie), and so on. In all these solutions, therefore, there are regions of 
both outward and inward flow (though of course the total discharge Q 
is positive). As R -> oo the number of alternating minima and maxima 
increases without limit, so that there is no definite limiting solution. We 
may emphasise that in divergent flow as R-> oo the solution does not, 
therefore, tend to the solution of Euler's equations as it does for convergent 
flow. Finally, it may be mentioned that, as R increases, the steady divergent 
flow of the kind described becomes unstable soon after R exceeds R m ax> 
and in practice a non-steady or turbulent flow occurs (Chapter III). 

(3) Determine the flow in a jet emerging from the end of a narrow tube 
into an infinite space filled with the fluid — the submerged jet (L. Landau, 
1943). 

We take spherical co-ordinates r, 6, <f>, with the polar axis in the direction 
of the jet at its point of emergence, and with this point as origin. The flow is 
symmetrical about the polar axis, so that v^ = and v dl v r are functions of r 
and 6 only. The same total momentum flux (the "momentum of the jet") 
must pass through any closed surface surrounding the origin (in particular, 
through an infinitely distant surface). For this to be so, the velocity must be 
inversely proportional to r, so that 

v r = F(d)fr, v e = f(d)[r, (23.16) 

where F and / are some functions of 6 only. The equation of continuity is 

1 d{r*v r ) 1 d 

+ — (v g sin 6) = 0. 

r 2 dr r sin 6 88 

Hence we find that 

F(d) = -dfldd-fcot9. (23.17) 

The components II r ^, 11^ of the momentum flux density tensor in the jet 
vanish identically by symmetry. We assume that the components 11^ 
and 11^ also vanish; this assumption is justified when we obtain a solution 
satisfying all the necessary conditions. Using the expressions (15.17) for 
the components of the tensor o%, and formulae (23.16), (23.17), we easily 
see that the relation 

sin2 lire = - ^[sin2 0(II^-rU)] 

holds between the components of the momentum flux density tensor in the 



§23 Exact solutions of the equations of motion for a viscous fluid 87 

jet. Hence it follows that U r e = 0. Thus only the component II rr is non- 
zero, and it varies as 1/r 2 . It is easy to see that the equations of motion 
dUijcjdXk = are automatically satisfied. 
Next, we write 

(n*-n*)/p = (p+2vfcote-2vf)ir2 = o, 

or 

d(l//)/d0+(l//)cot0+l/2i> = 0. 
The solution of this equation is 

/= -2i/sin0/(^-cos0), (23.18) 

and then we have from (23.17) 

[ A 2 - 1 \ 

F = 2v 1 . (23.19) 

l(,4-cos0)2 / v ' 

The pressure distribution is found from the equation 

IWP = Plp+f(f+2vcote)lr* = 0, 

which gives 

4pv 2 (^cos0-l) 

r z (A — cos 0) 2 

The constant A can be found in terms of the momentum of the jet, i.e. the 
total momentum flux in it. This flux is equal to the integral over the surface 
of a sphere 

n 

P = <J> n„. cosfld/ = 2tt f r^Urr cos0sin0 d0. 



The value of II rr is given by 

( (A*-l)* A \ 

{ (A-cosd)* ~ A-cosdy 



1 4^2 ( (A 2 - 1) 2 A 



p r 2 l(^4-cos0)4 ^4-cos0f 

and a calculation of the integral gives 

P =16 ^( 1+ _J__^ log ^±ij. (23 . 21) 

Formulae (23.16)-(23.21) give the solution of the problem.f 



t The solution here obtained is exact for a jet regarded as emerging from a point source. If the 
finite dimensions of the tube mouth are taken into account, the solution becomes the first term of an 
expansion in powers of the ratio of these dimensions to the distance r from the mouth of the tube. 
This is why, if we calculate from the above solution the total mass flux through a closed surface sur- 
rounding the origin, the result is zero. A non-zero total mass flux is obtained when further teuns 
in the above-mentioned expansion are considered; see Yu. B. Rumer, Prikladnaya matematika i 
mekhanika 16, 255, 1952. 

The submerged laminar jet with a non-zero angular momentum has bee ndiscussed by L. G. 
LoItsyanskiI (ibid. 17, 3, 1953). 



Viscous Fluids 



§24 



The streamlines are determined by the equation drfv r = rd0/v e , integration 
of which gives r sin 2 6j{A — cos 6) = constant. Fig. 12 shows the streamlines 
in the jet (for A > 1). 




Fig. 12 

Let us consider two limiting cases, a weak jet (small momentum P) and a 
strong jet (large P). As P -> 0, the constant A tends to infinity: from (23.21) 
we have P = l&n-v 2 pjA. For the velocity in this case we have 

v d = —P sin 6 j&TTvpr, v r = P cos QjAm>pr. 

As P -> oo (strong jetf), A tends to unity: (23.21) gives A — l + |a 2 , where 
a = 32ttv 2 p/3P. For large angles (0 ~ 1), the velocity is given by 

v 6 = — (2v/r) cotffl, v r = —2v/r, 

but for small angles (0 ~ a) we have 

^ = _4^/( a 2 + ^8), » r = 8va 2 /(a 2 +0 2 ) 2 . 



§24. Oscillatory motion in a viscous fluid 

When a solid body immersed in a viscous fluid oscillates, the flow thereby 
set up has a number of characteristic properties. In order to study these, 
it is convenient to begin with a simple but typical example. Let us suppose 
that an incompressible fluid is bounded by an infinite plane surface which 
executes a simple harmonic oscillation in its own plane, with frequency w. 
We require the resulting motion of the fluid. We take the solid surface as the 
yz-iplane, and the fluid region as x > ; the jy-axis is taken in the direction 
of the oscillation. The velocity u of the oscillating surface is a function of 
time, of the form A cos (atf + <x). It is convenient to write this as the real 
part of a complex quantity: 

u = re(uoe~ ia)t ), 

where the constant uq = Ae- i0L is in general complex, but can always be made 
real by a proper choice of the origin of time. 



t However, it must be borne in mind that the flow in a sufficiently strong jet is actually turbulent 
(§35). 



§24 Oscillatory motion in a viscous fluid 89 

So long as the calculations involve only linear operations on the velocity 
k, we may omit the sign re and proceed as if u were complex, taking the real 
part of the final result. Thus we write 

Uy = u = uo e"K (24.1) 

The fluid velocity must satisfy the boundary condition v = u for x = 0, 
i.e. v x = v 2 = 0, v y = u. 

It is evident from symmetry that all quantities will depend only on the 
co-ordinate x and the time t . From the equation of continuity div v = 
we therefore have dv x /8x = 0, whence v x = constant = zero, from the 
boundary condition. Since all quantities are independent of the co-ordinates 
y and z, we have (v«grad)v = v x dyjdx, and since v x is zero it follows that 
(v«grad)v = identically. The equation of motion (15.7) becomes 

dv/dt = -(lj P )gradp + vAv. (24.2) 

This is a linear equation. Its ^-component is dpjdx = 0, i.e. p = constant. 
It is further evident from symmetry that the velocity v is everywhere in 
the ^-direction. For v y = v we have by (24.2) 

dvjdt = vd^vjdx 2 , (24.3) 

that is, a (one-dimensional) heat conduction equation. We shall look for a 
solution of this equation which is periodic in x and t, of the form 

with a complex amplitude Mo, so that v = u for x = 0. Substituting in 
(24.3), we find toy = vk 2 , whence 

k = V( ioi l v ) = ± (*'+ WW 2 *)* 
so that the velocity v is 

v _ u e -V(M/2v)z e iW(u/2v)x-o)f}- (24.4) 

we have taken k to have a positive imaginary part, since otherwise the velocity 
would increase without limit in the interior of the fluid, which is physically 
impossible. 

The solution obtained represents a transverse wave: its velocity v y — v 
is perpendicular to the direction of propagation. The most important pro- 
perty of this wave is that it is rapidly damped in the interior of the fluid : 
the amplitude decreases exponentially as the distance x from the solid 
surface increases. f 

Thus transverse waves can occur in a viscous fluid, but they are rapidly 
damped as we move away from the solid surface whose motion generates the 
waves. 

The distance S over which the amplitude falls off by a factor of e is called 
the depth of penetration of the wave. We see from (24.4) that 

S = V(2v/w). (24.5) 

t Over a distance of one wavelength the amplitude diminishes by a factor of c 2 " - K 540. 



90 Viscous Fluids §24 

The depth of penetration therefore diminishes with increasing frequency, but 
increases with the kinematic viscosity of the fluid. 

Let us calculate the frictional force acting on unit area of the plane oscil- 
lating in the viscous fluid. This force is evidently in the ^-direction, and is 
equal to the component a xy = v\dv y \dx of the stress tensor; the value of the 
derivative must be taken at the surface itself, i.e. at x = 0. Substituting 
(24.4), we obtain 

°xy = Vd^pX*- !)«• (24.6) 

Supposing Mo real and taking the real part of (24.6), we have 

<*xy = —-\/(co7]p)uoCos(cot + l7r). 

The velocity of the oscillating surface, however, is u = uq cos cot. There 
is therefore a phase difference between the velocity and the frictional force.f 
It is easy to calculate also the (time) average of the energy dissipation 
in the above problem. This may be done by means of the general formula 
(16.3); in this particular case, however, it is simpler to calculate the required 
dissipation directly as the work done by the frictional forces. The energy 
dissipated per unit time per unit area of the oscillating plane is equal to the 
mean value of the product of the force a xy and the velocity u y = u: 

- o zy u = i«o 2 \/(£ OM ?/>)- (24.7) 

It is proportional to the square root of the frequency of the oscillations, 
and to the square root of the viscosity. 

An explicit solution can also be given of the problem of a fluid set in 
motion by a plane surface moving in its plane according to any law u = u(t). 
We shall not pause to give the corresponding calculations here, since the 
required solution of equation (24.3) is formally identical with that of an 
analogous problem in the theory of thermal conduction, which we shall 
discuss in §52 (the solution is formula (52.15)). In particular, the frictional 
force on unit area of the surface is given by 



°xy 



yrjp r du(r) dr 
7J^^) : (24 ' 8) 

cf. (52.16). 



t For oscillations of a half -plane (parallel to its edge) there is an additional frictional force due to 
edge effects. The problem of the motion of a viscous fluid caused by oscillations of a half-plane, and 
also the more general problem of the oscillations of a wedge of any angle, can be solved by a class of 
solutions of the equation A/+& 2 / *= 0, used by A. Sommehfeld in the theory of diffraction by a wedge; 
see, for instance, M. von Laue, Interferenz und Beugung elektromagnetischer Wellen (Interference 
and diffraction of electromagnetic waves), Handbuch der Experimentalphysik 18, 333, Akademische 
Verlagsgesellschaft, Leipzig 1928. 

We give here, for reference, only one result: the increase in the frictional force on a half-plane, 
arising from the edge effect, can be regarded as the result of increasing the area of the half-plane by 
moving the edge a distance JS = -\Z(vj2to). 



§24 Oscillatory motion in a viscous fluid 91 

Let us now consider the general case of an oscillating body of arbitrary 
shape. In the case of an oscillating plane considered above, the term 
(v»grad)v in the equation of motion of the fluid was identically zero. This 
does not happen, of course, for a surface of arbitrary shape. We shall assume, 
however, that this term is small in comparison with the other terms, so that 
it may be neglected. The conditions necessary for this procedure to be valid 
will be examined below. 

We shall therefore begin, as before, from the linear equation (24.2). 
We take the curl of both sides; the term curlgradp vanishes identically, 
giving 

2(curlv)/3* = vAcurlv, (24.9) 

i.e. curl v satisfies a heat conduction equation. We have seen above, however, 
that such an equation gives an exponential decrease of the quantity which 
satisfies it. We can therefore say that the vorticity decreases towards the 
interior of the fluid. In other words, the motion of the fluid caused by the 
oscillations of the body is rotational in a certain layer round the body, while 
at larger distances it rapidly changes to potential flow. The depth of penetra- 
tion of the rotational flow is of the order of 8 ~ -^(vjcj). 

Two important limiting cases are possible here: the quantity 8 may be 
either large or small compared with the dimension of the oscillating body. 
Let / be the order of magnitude of this dimension. We first consider the case 
8 > /; this implies that 1 2 oj <^ v. Besides this condition, we shall also suppose 
that the Reynolds number is small. If a is the amplitude of the oscillations, 
the velocity of the body is of the order of aco. The Reynolds number for the 
motion in question is therefore cual/v. We therefore suppose that 

Pa> <^ v, waljv < 1. (24.10) 

This is the case of low frequencies of oscillation, which in turn means that 
the velocity varies only slowly with time, and therefore that we can neglect 
the derivative dvfdt in the general equation of motion. The term (v»grad)v, 
on the other hand, can be neglected because the Reynolds number is small. 

The absence of the term dv]dt from the equation of motion means that the 
flow is steady. Thus, for 8 > /, the flow can be regarded as steady at any 
given instant. This means that the flow at any given instant is what it would 
be if the body were moving uniformly with its instantaneous velocity. If, 
for example, we are considering the oscillations of a sphere immersed in the 
fluid, with a frequency satisfying the inequalities (24.10) (/ being now the 
radius of the sphere), then we can say that the drag on the sphere will be that 
given by Stokes' formula (20.14) for uniform motion of the sphere at small 
Reynolds numbers. 

Let us now consider the opposite case, where / > 8. In order that the 
term (v«grad)v should again be negligible, it is necessary that the amplitude 
of the oscillations should be small in comparison with the dimensions of the 
body: 

Pco >v, a<$l; (24.11) 



92 Viscous Fluids §24 

in this case, it should be noticed, the Reynolds number need not be small. 
The above inequality is obtained by estimating the magnitude of (v«grad)v. 
The operator (v»grad) denotes differentiation in the direction of the velocity. 
Near the surface of the body, however, the velocity is nearly tangential. In 
the tangential direction the velocity changes appreciably only over distances 
of the order of the dimension of the body. Hence 

(v-grad)v ~ v 2 jl ~ a 2 a> 2 /l, 

since the velocity itself is of the order of am. The derivative d\jdt, however, 
is of the order of vco ~ aco 2 . Comparing these, we see that 

(v^grad)v 4 d\\dt 

if a < /. The terms dvjdt and v A v are then easily seen to be of the same 
order. 

We may now discuss the nature of the flow round an oscillating body when 
the conditions (24.11) hold. In a thin layer near the surface of the body 
the flow is rotational, but in the rest of the fluid we have potential flow.f 
Hence the flow everywhere except in the layer adjoining the body is given 
by the equations 

curlv = 0, div v = 0. (24.12) 

Hence it follows that Av = 0, and the Navier-Stokes equation reduces to 
Euler's equation. The flow is therefore ideal everywhere except in the 
surface layer. Since this layer is thin, in solving equations (24.12) to deter- 
mine the flow of the rest of the fluid we should take as boundary conditions 
those which must be satisfied at the surface of the body, i.e. that the fluid 
velocity is equal to that of the body. The solutions of the equations of motion 
for an ideal fluid cannot satisfy these conditions, however. We can require 
only the fulfilment of the corresponding condition for the fluid velocity 
component normal to the surface. 

Although equations (24.12) are inapplicable in the surface layer of fluid, 
the velocity distribution obtained by solving them satisfies the necessary 
boundary condition for the normal velocity component, and the actual 
variation of this component near the surface therefore has no significant 
properties. The tangential component would be found, by solving the equa- 
tions (24.12), to have some value different from the corresponding velocity 
component of the body, whereas these velocity components should be equal 
also. Hence the tangential velocity component must change rapidly in the 
surface layer. The nature of this variation is easily determined. Let us 
consider any portion of the surface of the body, of dimension large compared 



f For oscillations of a plane surface not only curl v but also v itself decreases exponentially with 
characteristic distance 8. This is because the oscillating plane does not displace the fluid, and there- 
fore the fluid remote from it remains at rest. For oscillations of bodies of other shapes the fluid is 
displaced, and therefore executes a motion where the velocity decreases appreciably only over distances 
of the order of the dimension of the body. 



§24 Oscillatory motion in a viscous fluid 93 

with S, but small compared with the dimension of the body. Such a portion 
may be regarded as approximately plane, and therefore we can use the re- 
sults obtained above for a plane surface. Let the #-axis be directed along 
the normal to the portion considered, and the ^y-axis parallel to the tangential 
velocity component of the surface there. We denote by v y the tangential 
component of the fluid velocity relative to the body; v y must vanish on the 
surface. Lastly, let voe~ ia)t be the value of v y found by solving equations 
(24.12). From the results obtained at the beginning of this section, we can 
say that in the surface layer the quantity v y will fall off towards the surface 
according to the law 

Vy = VQ e-t»4[l-er0--O*vW2r)] m (24.13) 

Finally, the total amount of energy dissipated in unit time will be given by 
the integral 

ikin = -W(blP°>) j> bo| 2 d/ (24.14) 

taken over the surface of the oscillating body. 

In the Problems at the end of this section we calculate the drag on various 
bodies oscillating in a viscous fluid. Here we shall make the following general 
remark regarding these forces. Writing the velocity of the body in the complex 
form u = uoe- iwt , we obtain a drag F proportional to the velocity u, and also 
complex: F = /?«, where f3= fii + ife is a complex constant. This expression 
can be written as the sum of two terms with real coefficients : 

F = (pi + ife)u = fau-feujco, (24.15) 

one proportional to the velocity u and the other to the acceleration u. 

The (time) average of the energy dissipation is given by the mean product 
of the drag and the velocity, where of course we must first take the real 
parts of the expressions given above, i.e. u = \{uoe~ i<at + uo*e io>t ), 
F = l{uofie- iwt + uo*p*e i0it ). Noticing that the mean values of e ±Zia)t are 
zero, we have 

Fu = l(P+P>) H 2 = ifr M 2 - (24.16) 

Thus we see that the energy dissipation arises only from the real part of /?; 
the corresponding part of the drag (24.15), proportional to the velocity, may 
be called the dissipative part. The other part of the drag, proportional to 
the acceleration and determined by the imaginary part of /?, does not involve 
the dissipation of energy and may be called the inertial part. 

Similar considerations hold for the moment of the forces on a body execut- 
ing rotary oscillations in a viscous fluid. 

PROBLEMS 

Problem 1. Determine the frictional force on each of two parallel solid planes, between 
which is a layer of viscous fluid, when one of the planes oscillates in its own plane. 



94 Viscous Fluids §24 

Solution. We seek a solution of equation (24.3) in the formf 
v = (A sinkx+B coskx)e~ i<ot , 

and determine A and B from the conditions v = u = M e~ i& " for * = and v — for 
x — h, where h is the distance between the planes. The result is 

sin k(h — x) 
V = u- 



sinkh 
The frictional force per unit area on the moving plane is 

Plx = rj(dv/dx)x=0 = —rjkucotkh, 
while that on the fixed plane is 

P<l x = — 7}(8vldx) x =h = f\hi cosec kh, 

the real parts of all quantities being understood. 

Problem 2. Determine the frictional force on an oscillating plane covered by a layer of 
fluid of thickness h, the upper surface being free. 

Solution. The boundary condition at the solid plane is v = u for * ■= 0, and that at the 
free surface is a xy = rjdv/dx = for x = h. We find the velocity 

COS k(h — x) 

V = u . 

coskh 

The frictional force is 

P x = 7)(dvjdx) x =^o — yku tan kh. 

Problem 3. A plane disk of large radius R executes rotary oscillations of small amplitude 
about its axis, the angle of rotation being = 8 cos <at, where & <^ 1 . Determine the moment 
of the frictional forces acting on the disk. 

Solution. For oscillations of small amplitude the term (v.grad)v in the equation of 
motion is always small compared with 8v/8t, whatever the frequency to. If R ^> 8, the disk 
may be regarded as infinite in determining the velocity distribution. We take cylindrical 
co-ordinates, with the jar-axis along the axis of rotation, and seek a solution such that 
Vr = Vt = 0, v* = v = rQ(s, t). For the angular velocity Q(z, t) of the fluid we obtain the 
equation 

BQjdt = vdm/dz 2 . 
The solution of this equation which is — w6 Q sin tot for z = and zero for z = oo is 

Q = - <od e~ z/s sin(wt - z/8). 

The moment of the frictional forces on both sides of the disk is 

R 
M = 2 f r'27rrr)(dvldz) s= >odr = wOvn^/(u)prq)R* cos(arf-iir). 



t In all the Problems to this section S denotes the quantity (24.5): 

8 = V( 2v h)> and k = (l + i)IS. 



§24 Oscillatory motion in a viscous fluid 95 

Problem 4. Determine the flow between two parallel planes when there is a pressure 
gradient which varies harmonically with time. 

Solution. We take the war-plane half-way between the two planes, with the w-axis parallel 
to the pressure gradient, which we write in the form 

-(l/p)dp/dx = ae~ i(at . 

The velocity is everywhere in the ^-direction, and is determined by the equation 

dv/dt = ae-M+vdHldy*. 

The solution of this equation which satisfies the conditions v — for y = ±JA is 

COS Ay 
COS^M. 
The mean value of the velocity over a cross-section is 

2 



ia r cos Ay 1 

v = — e- itot \ 1 — . 

co I cosiMj 



ia l I \ 

v = — e~ ia>t 1 tan AM . 

co \ kh * / 



For h/8 <^ 1 this becomes 

v « ae-t^h 2 1 12v, 

in agreement with (17.5), while for hjh ^> 1 we have 

v x (ia/co)e~ i<at , 

in accordance with the fact that in this case the velocity must be almost constant over the 
cross-section, varying only in a narrow surface layer. 

Problem 5. Determine the drag on a sphere of radius R which executes translatory oscil- 
lations in a fluid. 

Solution. We write the velocity of the sphere in the form u = u c _ * <u '. As in §20, we 
seek the fluid velocity in the form v = e~ ib)t curl curl/u , where / is a function of r only 
(the origin is taken at the instantaneous position of the centre of the sphere). Substituting in 
(24.9) and effecting transformations similar to those of §20, we obtain the equation 

A a /+(M>W A/=0 

(instead of the equation A 2 / = in §20). Hence we have 

A/ = constant x e ikr /r, 

the solution being chosen which decreases exponentially with r. Integrating, we have 

df[dr = [ae^r(r-lfik) + b]lr^; (1) 

the function/ itself is not needed, since only the derivatives/ ' and/ " appear in the velocity. 
The constants a and b are determined from the condition that v = u for r = R, and are 
found to be 

3R / 3 3 \ 

a = - —e-i**, b = -W[\ . (2) 

lik ¥ \ ikR k*R*I K) 

It may be pointed out that, at large distances {R ^> 8), a -> and b -> — \R* t the values for 
potential flow obtained in §10, Problem 2; this is in accordance with what was said in §24. 



96 Viscous Fluids §24 

The drag is calculated from formula (20.13), in which the integration is over the surface 
of the sphere. The result is 

/ R \ I 2R \ du 

F = 6^1 + - )u + 3iTRW(2rip/co)(l + — J — . (3) 

For to = this becomes Stokes' formula, while for large frequencies we have 

du 

F = %tt P R* + 2>TTR^(2r }P oi)u. 

dt 

The first term in this expression corresponds to the inertial force in potential flow past a 
sphere (see §11, Problem 1), while the second gives the limit of the dissipative force. 

Problem 6. Determine the drag on a sphere moving in an arbitrary manner, the velocity 
being, given by a function u{t). 

Solution. We represent u(t) as a Fourier integral : 

f ! f 

M (0 = u^e-^tdco, u u = — u{r)e ib>T dr. 

J 2tt J 

— oo —oo 

Since the equations are linear, the total drag may be written as the integral of the drag forces 
for velocities which are the separate Fourier components u (t) e~ iwt ] these forces are given by 
(3) of Problem 5, and are 

Noticing that (du/dt)^ = —icou^, we can rewrite this as 

( 6v 3V(2v) 1 + M 

On integration over to, the first and second terms give respectively u(t) and u(t). To integrate 
the third term, we notice first of all that for negative co this term must be written in the 
complex conjugate form, (1 +i)j\/co being replaced by (1 — *)/ 'VMi tm s is because formula 
(3) of Problem 5 was derived for a velocity u — u e~ i<Jt with to > 0, and for a velocity 
«„£*«' we should obtain the complex conjugate. Instead of an integral over w from — oo 
to + oo, we can therefore take twice the real part of the integral from to oo. We write 



oo, . , . oo oo 



2iel(l+i) K -^ da) = -re (1 + -^- dwdr 

-oo 

oo oo oo 

1 i r r u(t) e- ib *t-r) r /• u (t) eft**-** 

--re (1+0 -^- dcodr + (l + -^ dcodi 

IT \ J J \/CO J J -S/Oi 

-oo * 



t 00 

Vtt \ J V(t-r) J V(t-0 J 



N IT J 



—oo 
t 



U(r) 



V(t-r-\ 



-dr. 



§24 Oscillatory motion in a viscous fluid 97 

Thus we have finally for the drag 

i\du 3vu 3 iv rdu dr ) 

f=2lrpi? 3{__ + _ + _y-j_ j. (1) 

— oo 

Problem 7. Determine the drag on a sphere which at time t = begins to move with a 
uniform acceleration, u = at. 

Solution. Putting, in formula (1) of Problem 6, u = for t < and u = at for t > 
we have for t > 



rl 3vt 6 /tvi 
F = 2.pfi3«[- + - + - A /-J. 



Problem 8. The same as Problem 7, but for a sphere brought instantaneously into uniform 
motion. 

Solution. We have u = f or t < and u = u for t > 0. The derivative dujdt is 
zero except at the instant t = 0, when it is infinite, but the time integral of du/dt is finite, 
and equals « . As a result, we have for all t > 

[D -i 

1 + — - — - +&rpR 3 U 8(t), 
V(irrt) J 

where B(t) is the delta function. For t -> oo this expression tends asymptotically to the value 
given by Stokes' formula. The impulsive drag on the sphere at t = is obtained by integrat- 
ing the last term and is %irpR 3 u . 

Problem 9. Determine the moment of the forces on a sphere executing rotary oscillations 
about a diameter in a viscous fluid. 

Solution. For the same reasons as in §20, Problem 1, the pressure-gradient term can be 
omitted from the equation of motion, so that we have dv/dt = v A v. We seek a solution in 
the form v = curl/£2 <r i6 ' t > where SI = Sl e~ ib)t is the angular velocity of rotation of the 
sphere. We then obtain for/, instead of the equation A/ = constant, 

A/+ k 2 f — constant. 

Omitting an unimportant constant term in the solution of this equation, we find/ = ae ikr lr 
taking the solution which vanishes at infinity. The constant a is determined from the boundary 
condition that v = SI X r at the surface of the sphere. The result is 

i? 3 / R\ 3 l-ikr ,, x 

/ = e ik(r-R) v = (SI xr) — z — ^ Wr_iJ) , 

where R is the radius of the sphere. A calculation like that in §20, Problem 1, gives the fol- 
lowing expression for the moment of the forces exerted on the sphere by the fluid: 

&r 3 + 6R/8+ 6(K/8)2+ 2(i?/3)s- 2«(R/S)2(1 +JJ/8) 
M ~ ~ T^ l + 2*/8 + 2(W ' 

For co -> (i.e. S -> oo), we obtain M = — 8itt)R 3 Q, corresponding to uniform rotation 
of the sphere (see §20, Problem 1). In the opposite limiting case RI8 > 1, we find 

4a/2 
M = -^--ir2?V0v>G>)(* - 1) Q - 



98 Viscous Fluids §25 

This expression can also be obtained directly: for S <^ R each element of the surface of the 
sphere may be regarded as plane, and the frictional force acting on it is found by substituting 
a = OR sin in formula (24.6). 

Problem 10. Determine the moment of the forces on a hollow sphere filled with viscous 
fluid and executing rotary oscillations about a diameter. 

Solution. We seek the velocity in the same form as in Problem 9. For/ we take the solu- 
tion (a/r) sin kr, which is finite everywhere within the sphere, including the centre. Deter- 
mining a from the boundary condition, we have 

R \ 3 krcoskr—sinkr 
v = (Hxr)l- 



= (ftxr)(- 



/ kR cos kR — sin kR ' 

A calculation of the moment of the frictional forces gives the expression 

k 2 R* sin kR + 3kR cos kR - 3 sin kR 



M = l7rr)R 3 Q- 



kRcoskR—sinkR 



The limiting value for S > 1 is of course the same as in the preceding problem. If 
R/S <^1 we have 

R2co 



I R?co \ 



The first term corresponds to the inertial forces occurring in the rigid rotation of the whole 
fluid. 



§25. Damping of gravity waves 

Arguments similar to those given above can be advanced concerning the 
velocity distribution near the free surface of a fluid. Let us consider oscil- 
latory motion occurring near the surface (for example, gravity waves). 
We suppose that the conditions (24.11) hold, the dimension / being now re- 
placed by the wavelength A: 

A2w > v , a<£\; (25.1) 

a is the amplitude of the wave, and w its frequency. Then we can say that 
the flow is rotational only in a thin surface layer, while throughout the rest 
of the fluid we have potential flow, just as we should for an ideal fluid. 

The motion of a viscous fluid must satisfy the boundary conditions (15.14) 
at the free surface; these require that certain combinations of the space 
derivatives of the velocity should vanish. The flow obtained by solving the 
equations of ideal-fluid dynamics does not satisfy these conditions, however. 
As in the discussion of % in the previous section, we may conclude that the 
corresponding velocity derivatives decrease rapidly in a thin surface layer. 
It is important to notice that this does not imply a large velocity gradient as 
it does near a solid surface. 

Let us calculate the energy dissipation in a gravity wave. Here we must 
consider the dissipation, not of the kinetic energy alone, but of the mechanical 
energy E mech , which includes both the kinetic energy and the potential 



§25 Damping of gravity waves 99 

energy in the gravitational field. It is clear, however, that the presence or 
absence of a gravitational field cannot affect the .energy dissipation due to 
processes of internal friction in the fluid. Hence E mech is given by the same 
formula (16.3): 

J \ OXjc OXi l 

In calculating this integral for a gravity wave, it is to be noticed that, since 
the volume of the surface region of rotational flow is small, while the velocity 
gradient there is not large, the existence of this region may be ignored, unlike 
what was possible for oscillations of a solid surface. In other words, the inte- 
gration is to be taken over the whole volume of fluid, which, as we have seen, 
moves as if it were an ideal fluid. 

The flow in a gravity wave for an ideal fluid, however, has already been 
determined in §12. Since we have potential flow, 

dvijdxk = d 2 <f>/dxjcdxi = Bvjt/dxu 
so that 



A— -a* J '(^k) dV - 



The potential <f> is of the form 

<j> = <f>ocos(kx—o)t + (x.)e~ kz . 

We are interested, of course, not in the instantaneous value of the energy 
dissipation, but in its mean value £ mech with respect to time. Noticing that 
the mean values of the squared sine and cosine are the same, we find 



•Cmech 



= -8^4 (<jfidV. (25.2) 



The energy E mech itself may be calculated for a gravity wave by using a 
theorem of mechanics that, in any system executing small oscillations (of 
small amplitude, that is), the mean kinetic and potential energies are equal. 
We can therefore write E meeh simply as twice the kinetic energy: 

£ me ch = P jv*dV = pj (d<f>ldxi)2 dV, 

whence 

Em** = 2p& JP dV. (25.3) 

The damping of the waves is conveniently characterised by the damping 
coefficient y, defined as 

y = | E m ech \l2E meC h* (25.4) 



100 Viscous Fluids §25 

In the course of time, the energy of the wave decreases according to the law 
^mech = constant x e~ 2n ; since the energy is proportional to the square of 
the amplitude, the latter decreases with time as e~ n . 
Using (25.2), (25.3), we find 

y = 2vk 2 . (25.5) 

Substituting here (12.7), we obtain the damping coefficient for gravity waves 
in the form 

y = 2vcD*lg*. (25.6) 

PROBLEMS 

Problem 1. Determine the damping coefficient for long gravity waves propagated in a 
channel of constant cross-section; the frequency is supposed so large that ■y/ivloS) is small 
compared with the depth of the fluid in the channel. 

Solution. The principal dissipation of energy occurs in the surface layer of fluid, where 
the velocity changes from zero at the boundary to the value v = v e~ im which it has in the 
wave. The mean energy dissipation per unit length of the channel is by (24.14) Hv^^/i^pcolS), 
where / is the perimeter of the part of the channel crossjsection occupied by the fluid. The 
mean energy of the fluid (again per unit length) is Spv 2 — iSp\vo\ a , where S is the cross- 
sectional area of the fluid in the channel. The damping coefficient is y = l-\Z(vcx)/SS 2 ). 
For a channel of rectangular section, therefore, 

2h + a 

where a is the width and h the depth of the fluid. 

Problem 2. Determine the flow in a gravity wave on a very viscous fluid. 

Solution. The calculation of the damping coefficient as shown above is valid only when 
this coefficient is small, so that the motion may be regarded as that of an ideal fluid to a first 
approximation. For arbitrary viscosity we seek a solution of the equations of motion 



/ d*V x d*V x \ 1 dp 
\ dx 2 dz 2 J p dx' 
dv z _ I 8 2 v z 8 2 v z \ I dp 
~dt ~ V [~8x^' + ~8z 2 ') "p^ - ^ 



dv x dv z 

— - + — -= 

dx dz 



(1) 



which depends on t and * as e- i(0t + ikx , and diminishes in the interior of the fluid (z < 0). 
We find 

ik 

v x = e - io)t+ikx (Ae kz +Be! mz ), v z = e~ M+ikx (-iAe kz Be mz ), 

m 

p/p = e -io>t+ikx coAe kz jk—gz, where m = ^(hP — ico/v). 
The boundary conditions at the fluid surface are 

(dv x dv z \ 
1 = for z = I. 
dz dx J 



§25 Damping of gravity waves 101 

In the second condition we can immediately put z = instead of z = £. The first condition, 
however, should be differentiated with respect to t, after which we replace gdlldt by gv z 
and then put z = 0. The condition that the resulting two homogeneous equations for A 
and B are compatible gives 

This equation gives w as a function of the wave number k; <o is complex, its real part giving 
the frequency of the oscillations and its imaginary part the damping coefficient. The solu- 
tions of equation (2) that have a physical meaning are those whose imaginary parts are nega- 
tive (corresponding to damping of the wave); only two roots of (2), meet this requirement. 
If vk 2 < V(g k ) ( the condition (25.1)), then the damping coefficient is small, and (2) gives 
approximately a> = ± V(gk)-i.2vk 2 , a result which we already know. In the opposite limit- 
ing case vk 2 > V(ik), equation (2) has two purely imaginary roots, corresponding to damped 
aperiodic flow. One root is to = -igl2vk, while the other is much larger (of order vk 2 ), 
and therefore of no interest, since the corresponding motion is strongly damped. 



CHAPTER III 

TURBULENCE 

§26. Stability of steady flow 

In solving the equations of steady flow for a viscous fluid, it is often necessary 
to make certain approximations on account of mathematical difficulties. 
The validity of these approximate solutions is, of course, restricted, Such, 
for instance, is the solution of the problem of flow past a sphere given in 
§20, which is valid only for small Reynolds numbers. 

In principle, however, there must be an exact stationary solution of the 
equations of fluid dynamics for any problem with given steady external 
conditions; such exact solutions have been considered in §§17, 18 and 23. 
These solutions formally hold for all Reynolds numbers. 

Yet not every solution of the equations of motion, even if it is exact, 
can actually occur in Nature. The flows that occur in Nature must not only 
obey the equations of fluid dynamics, but also be stable. For the flow to be 
stable it is necessary that small perturbations, if they arise, should decrease 
with time. If, on the contrary, the small perturbations which inevitably occur 
in the flow tend to increase with time, then the flow is absolutely unstable. 
Such a flow unstable with respect to infinitely small perturbations cannot 
exist. 

The mathematical investigation of the stability of a given flow with respect 
to infinitely small perturbations will proceed as follows. On the steady 
solution concerned (whose velocity distribution is v (x,y y z), say), we 
superpose a non-steady small perturbation vi (x, y, z, t), which must be 
such that the resulting velocity v = v + vi satisfies the equations of motion. 
The equation for vi is obtained by substituting in the equations 

Sv firad p 

— + (v-grad)v = + „Av, divv = 

ot p 

the velocity and pressure v = v + vi, p = p +p h where the known functions 
vo and po satisfy the unperturbed equations 

grad/>o 

(vo-grad)vo = + i>Av , divv = 0. 

P 
Omitting terms above the first order in vi, we obtain 

dvi 

+ (v • grad)vi + (vi • grad)vo 



8t 

gradpi 



+ vAvi, divvi = 0. (26.1) 



P 

The boundary condition is that vi vanishes on fixed solid surfaces. 

102 



§27 The onset of turbulence 103 

Thus vi satisfies a system of linear differential equations, with coefficients 
that are functions of the co-ordinates only, and not of the time. The general 
solution of such equations can be represented as a sum of particular solutions 
in which vi depends on time as e~ ia,t . The "frequencies" to of the perturba- 
tions are not arbitrary, but are determined by solving the equations (26.1) 
with the appropriate boundary conditions. The "frequencies" are in general 
complex. If there are w whose imaginary parts are positive, e~ i0it will 
increase indefinitely with time. In other words, such perturbations, once 
having arisen, will increase, i.e. the flow is unstable with respect to such 
perturbations. For the flow to be stable it is necessary that the imaginary 
part of any possible "frequency" a> is negative. The perturbations that arise 
will then decrease exponentially with time. 

Such a mathematical investigation of stability is extremely complicated, 
however. The theoretical problem of the stability of steady flow past bodies 
of finite dimensions has not yet been solved. It is certain that steady flow is 
stable for sufficiently small Reynolds numbers. The experimental data 
seem to indicate that, when R increases, it eventually reaches a value Rcr 
(the critical Reynolds number) beyond which the flow is unstable with respect 
to infinitesimal disturbances. For sufficiently large Reynolds numbers 
(R > Rcr), steady flow past solid bodies is therefore impossible. The 
critical Reynolds number is not, of course, a universal constant, but takes a 
different value for each type of flow. These values appear to be of the order 
of 10 to 100; for example, in flow across a cylinder undamped non-steady 
flow has been observed for R = udjv = 34, d being the diameter of the 
cylinder. Exact measurements of R C r, however, have not been made. 

§27. The onset of turbulence 

Let us now consider the nature of the non-steady flow which is established 
as a result of the absolute instability of steady flow at large Reynolds numbers. 
We begin by examining the properties of this flow at Reynolds numbers only 
slightly greater than R cr . For R < R cr the imaginary parts of the complex 
"frequencies" <o = a>i + iyi for all possible small velocity perturbations are 
negative (yi < 0). For R = R cr there is one frequency whose imaginary part 
is zero. For R > R er the imaginary part of this frequency is positive, but, 
when R is close to R cr , yi is small in comparison with the real part wi.f 
The function vi corresponding to this frequency is of the form 

Vl = A(t)f(x,y,z), (27.1) 

where f is some complex function of the co-ordinates, and the complex 
"amplitude" A(t) is$ 

A(t) = constant KePter*^*. (27.2) 



t It must be borne in mind that the set (or spectrum) of all possible frequencies for a given type 
of flow includes both separate isolated values (the discrete spectrum) and the whole of various fre- 
quency ranges (the continuous spectrum). However, it can be seen that the frequencies with positive 
imaginary parts in which we are interested occur, in general, Only in the discrete spectrum. 

J As usual, we understand the real part of (27.2). 



104 Turbulence §27 

This expression for A(t) is actually valid, however, only during a short 
interval of time after the disruption of the steady flow; the factor e y ^ increases 
rapidly with time, whereas the method of determining vi given in §26, 
which leads to expressions like (27.1) and (27.2), applies only when jvij 
is small. In reality, of course, the modulus \A | of the amplitude of the non- 
steady flow does not increase without limit, but tends to a finite value. 
For R close to R C r (we always mean, of course, R > R cr ), this finite value is 
small, and can be determined as follows. 

Let us find the time derivative of the squared amplitude \A | 2 . For very 
small values of t, when (27.2) is still valid, we have d|^| 2 /di = 2yi|^| 2 . 
This expression is really just the first term in an expansion in series of powers 
of A and A*. As the modulus \A | increases (still remaining small), sub- 
sequent terms in this expansion must be taken into account. The next 
terms are those of the third order in A. However, we are not interested in 
the exact value of the derivative d|.4| 2 /d/, but in its time average, taken 
over times large compared with the period 2ttJo}\ of the factor g-^i*; we 
recall that, since coi > yi, this period is small compared with the time 1/yi 
required for the amplitude modulus \A | to change appreciably. The third- 
order terms, however, must contain the periodic factor, and therefore vanish 
on averaging.f The fourth-order terms include one which is proportional 
to A 2 A* 2 = \A | 4 and which clearly does not vanish on averaging. Thus we 
have as far as fourth-order terms 



d\Ap/dt = 2 n \A\*-K\A\*. (27.3) 

where a may be either positive or negative. 

Let us suppose that a is positive. J We have not put bars above \A\ 2 
and \A | 4 , since the averaging is only over time intervals short compared with 
1/yi. For the same reason, in solving the equation we proceed as if the bar 
were omitted above the derivative also. The solution of equation (27.3) is 

1/|^4| 2 = a/2yi + constant xr 2 M 
Hence it is clear that \A | 2 tends asymptotically to a finite limit: 

|^| 2 max = 2yi/a. (27.4) 

The quantity y\ is some function of the Reynolds number. Near R cr it 
can be expanded as a series of powers of R— R cr . But yi(R C r) = 0, by 
the definition of the critical Reynolds number. Hence the zero-order term 
in the expansion is zero, and we have to the first order y\ = constant x 
(R— R C r). Substituting this in (27.4), we see that the modulus \A\ of the 
amplitude is proportional to the square root of R— R cr : 

|^|max~ V(R-Rcr). (27.5) 



f Strictly speaking, the third-order terms give, on averaging, not zero, but fourth-order terms, 
which we suppose included among the fourth-order terms in the expansion. 
J This seems to be true for ordinary flow past bodies. 



§27 The onset of turbulence 105 

Let us summarise these results. The absolute instability of the flow for 
R > R cr leads to the appearance of a non-steady periodic flow. For R close 
to Rcr the latter flow can be represented by superposing on the steady flow 
vo (x, y, z) a periodic flow vi{x, y, z, t) y with a small but finite amplitude 
which increases with R proportionally to the square root of R-R er - The 
velocity distribution in this flow is of the form 

vi = f(*,v,*>-*KW, (27.6) 

where f is a complex function of the co-ordinates, and pi is some initial 
phase. For large R-R cr , the separation of the velocity into v and vi is 
no longer meaningful. We then have simply some periodic flow with fre- 
quency oji. If, instead of the time, we use as an independent variable the 
phase <£i s uit+pi, then we can say that the function v(#, y, *, <£i) is a 
periodic function of <£i, with period 2tt. This function, however, is no 
longer a simple trigonometrical function. Its expansion in Fourier series 

V 

(where the summation is over all integers p, positive and negative) includes 
not only terms with the fundamental frequency a>i, but also terms whose 
frequencies are integral multiples of coi. 

The following important property of this non-steady flow should also be 
mentioned. Equation (27.3) determines only the modulus of the time factor 
A(t), and not its phase. The phase <fc = ant + fii of the periodic flow remains 
essentially indeterminate, and depends on the particular initial conditions 
which happen to occur at the instant when the flow begins. The initial 
phase ft can have any value, depending on these conditions. Thus the 
periodic flow under consideration is not uniquely determined by the given 
steady external conditions in which the flow takes place. One quantity— the 
initial phase of the velocity— remains arbitrary. We may say that the flow 
has one degree of freedom, whereas steady flow, which is entirely determined 
by the external conditions, has no degrees of freedom. 

Let us now consider the phenomena which occur when the Reynolds 
number increases further. When this happens, a time finally comes when the 
periodic flow discussed above in turn becomes unstable. The investigation of 
this instability would proceedf similarly to the method given above for 
determining the instability of the original steady flow. The part of the un- 
perturbed flow is now taken by the periodic flow v (x, y, z, t) (with frequency 
wi), and in the equations of motion we substitute v = v + v 2 , where v 2 
is a small correction. For v 2 we again obtain a linear equation, but the co- 
efficients are now functions of time as well as of the co-ordinates, being 



t But has not been carried out even for particular cases, on account of the exceptional mathematical 
difficulties. 



*06 Turbulence 



§27 



periodic in time with period 2tt/coi. The solution of such an equation must 
be sought in the form v 2 = U(x, y, z, t)er^\ where II(tf, y, z f t) is a periodic 
function of time, with period 2tt/o> 1 . The instability again occurs when a fre- 
quency cu = io 2 + iy 2 appears such that the imaginary part y 2 is positive, and 
the corresponding real part o> 2 then determines the new frequency which 
appears. 

The result, therefore, is that a quasi-periodic flow appears, characterised 
by two different periods. Just as the flow had one degree of freedom after 
the appearance of the first periodic flow, so it now involves two arbitrary 
quantities (phases), i.e. it has two degrees of freedom. 

When the Reynolds number increases still further, more and more new 
periods appear in succession. The range of Reynolds numbers between 
successive appearances of new frequencies diminishes rapidly in size. The 
new flows themselves are on a smaller and smaller scale. This means that 
the order of magnitude of the distances over which the velocity changes 
appreciably is the smaller, the later the flow in question appears. 

For R > R cr , therefore, the flow rapidly becomes complicated and con- 
fused. Such a flow is said to be turbulent; its properties will be investigated 
in detail in the following sections. In contradistinction to turbulent flow, 
the regular flow, in which the fluid moves as it were in layers with different 
velocities, is said to be laminar. 

We can write down the general form of a function v(x, y, z, t) whose time 
dependence is given by some number n of different frequencies w } (j = 1, 
2, ..., «). Instead of one phase fa = a>i*+ft, we now have n different phases 
fa = wjt+Pj. The function v may be regarded as a function of these phases 
(and of the co-ordinates), and is periodic in each of them, with period 2tt. 
Such a function can be written as a series: 



v(x,y,z,t)= £ ^....Pni^yy^expl-if^p^l (27.8) 

Pi'Pv-Pn i=l 

the summation being taken over all integrals p h p 2 , ..., p n . This is a generali- 
sation of formula (27.7). We may notice that the choice of the fundamental 
frequencies o>i, ..., co n is, as we see from (27.8), itself not unique; we could 
equally well take any n independent linear combinations of co t with integral 
coefficients.f 

A flow described by a formula such as (27.8) has n degrees of freedom; 
it involves n arbitrary initial phases /fy. As the Reynolds number increases, 
both the number of frequencies and the number of degrees of freedom 
increase. In the limit as R tends to infinity, the number of degrees of free- 
dom also increases indefinitely. 



t These linear combinations must be such that from them we can form all possible numbers 
S Pity. It is easy to see that, for this to be so, the determinant of the transformation coefficients 
relating the old and new frequencies must be unity. 



§28 Stability of flow between rotating cylinders 107 

It must be borne in mind that, since the velocity is a periodic function of 
the phases, with period 2tt, the states whose phases differ only by an integral 
multiple of 2rr are physically indistinguishable. In other words, we can 
say that all the essentially different values of each phase lie in the range 
s% fa ^ 277. Let us consider any two phases ^i = <oit+pia.ndfa = o>2*+/?2. 
Suppose that, at some instant, fa has the value a. Then, by what we have 
just said, fa will have values equivalent to a at all instants t = (a - ft. + 27rr)/a>i, 
where r is any integer. At these instants the phase fa will have the values 

fa = a>2(a-jSi)/ft>i+j82 + 277TG)2/c<;i. 

The different frequencies are generally incommensurable, so that co 2 /o>i 
is an irrational number. If we reduce each value of fa to the range to 2tt 
by subtracting the appropriate integral multiple of 2tt, we therefore obtain, 
as r goes from to oo, values for fa which are arbitrarily close to any given 
number in that range. In other words, in the course of a sufficiently long time 
fa and fa will simultaneously be arbitrarily close to any given pair of values. 
The same is obviously true of all the phases. Thus turbulent motion has 
a certain quasi-periodic property: in the course of a sufficiently long time the 
fluid passes through states arbitrarily close to any given state, determined by 
any possible choice of simultaneous values of the phases fa. 

We have introduced the concept of the critical Reynolds number as being 
the value of R at which instability of steady flow, in the sense described above, 
first occurs. The critical Reynolds number can, however, be regarded from 
a somewhat different point of view. For R < R cr there are no stable non- 
steady solutions of the equations of motion that are not damped in time. 
After the critical value has been reached, a stable non-steady solution appears, 
which will actually occur in a moving fluid. 

As far as experimental investigations of the flow past ordinary finite 
bodies are concerned, the two definitions of R cr seem to be the same. Logi- 
cally, however, this need not be so, and cases could in principle occur where 
there are two different critical values: one above which non-steady flow 
can occur without being damped, and another above which steady flow 
becomes absolutely unstable. The second must obviously be greater than 
the first. However, since there is at present no indication that such cases of 
instability actually exist, we shall not pause to investigate them more closely.f 

§28. Stability of flow between rotating cylinders 

To investigate the stability of steady flow between two rotating cylinders 
(§18) in the limit of very large Reynolds numbers, we can use a simple method 
like that used in §4 to derive the condition for mechanical stability of a fluid 
at rest in a gravitational field (Rayleigh, 1916). The principle of the method 
is to consider any small element of the fluid and to suppose that this element 



t We are not here concerned with (e.g.) flow in a pipe, where the loss of stability has unusual 
properties (see §29). 



108 Turbulence §28 

is displaced from the path which it follows in the flow concerned. As a result 
of this displacement, forces appear which act on the displaced element. If 
the original flow is stable, these forces must tend to return the element to 
its original position. 

Each fluid element in the unperturbed flow moves in a circle r = constant 
about the axis of the cylinders. Let fju(r) = tnr 2 <j> be the angular momentum 
of an element of mass m, <j> being the angular velocity. The centrifugal 
force acting on it is y?\mr z \ this force is balanced by the radial pressure 
gradient in the rotating fluid. Let us now suppose that a fluid element at a 
distance r from the axis is slightly displaced from its path, being moved to 
a distance r > r from the axis. The angular momentum of the element 
remains equal to its original value fj. = /i(r ). The centrifugal force acting 
on the element in its new position is therefore /xo 2 /wr 3 . In order that the 
element should tend to return to its initial position, this force must be less 
than the equilibrium value /x 2 /wr 3 which is balanced by the pressure gradient 
at the distance r. Thus the necessary condition for stability is /x 2 -/x 2 > 0. 
Expanding /x(r) in powers of the positive difference r-r , we can write this 
condition in the form 

ndfi/dr > 0. (28.1) 

According to formula (18.3), the angular velocity <j> of the moving fluid 
particles is 

_ Q 2 R2 2 -niRi 2 (Qi-n 2 )Ri 2 R2 2 1 
R 2 2 ~Ri 2 + R 2 2 -Ri 2 r2* 

Calculating //, = mr 2 <f> and omitting factors which are certainly positive, 
we can write the condition (28.1) as 

(Q 2 #2 2 -ai#i 2 >£ > 0. (28.2) 

The angular velocity <f> varies monotonically from Qi on the inner cylinder 
to Q 2 on the outer cylinder. If the two cylinders rotate in opposite directions, 
i.e. if £li and Q2 have opposite signs, the function (/> changes sign between the 
cylinders, and its product with the constant number D 2 i2 2 2 -^ii?i 2 cannot 
be everywhere positive. Thus in this case (28.2) does not hold at all points 
in the fluid, and the flow is unstable. 

Now let the two cylinders be rotating in the same direction; taking this 
direction of rotation as positive, we have Qi > 0, Q 2 > 0. Then </> is every- 
where positive, and for the condition (28.2) to be fulfilled it is necessary that 

Q 2 i?2 2 > Q1.R1 2 . (28.3) 

If Q2R2 2 < Q.1R1 2 the flow is unstable. For example, if the outer cylinder is 
at rest (Q 2 = 0), while the inner one rotates, then the flow is unstable. If, 
on the other hand, the inner cylinder is at rest (Qi = 0), the flow is stable. 

It must be emphasised that no account has been taken, in the above argu- 
ments, of the effect of the viscous forces when the fluid element is displaced. 



§28 Stability of flow between rotating cylinders 109 

The method is therefore applicable only for small viscosities, i.e. for large R. 
To investigate the stability of the flow for any R, it is necessary to follow 
the general method, starting from equations (26.1) (G. I. Taylor, 1923). 
In the present case the unperturbed velocity distribution vo depends only on 
the (cylindrical) radial co-ordinate r, and not on the angle <f> or the axial 
co-ordinate z. Thus we have for the perturbation vi a system of linear 
equations with coefficients which contain neither the time nor the co-ordinates 
<f> and z. We may seek solutions of these equations in the form 

Vl _ e iVcz-<ot)f( r ) f (28.4) 

the direction of the vector f being arbitrary; this solution depends on z 
through the periodic factor e ikz , and the wave number k determines the 
periodicity of the perturbation in the z-direction. The possible frequencies 
co, obtained by solving the equations with the necessary boundary conditions 
in a plane perpendicular to the axis (vi = for r = Ri and r = R 2 ), will 
then be functions of k, involving R as a parameter: co = co(k, R). The 
point where instability appears is determined by the value of R for which the 
function y\ = im co first becomes zero for some k. For R < R C r, the func- 
tion yi(k, R) is always negative, but for R > R cr we have y 1 > in some range 
of k. Let k cr be the value of k for which yi = when R = R cr - The cor- 
responding function (28.4) gives the nature of the flow which occurs (super- 
posed on the original flow) in the fluid at the instant when the original flow 
ceases to be stable; it is periodic along the axis of the cylinders, with wave- 
length 277/& cr .t 

As well as solutions of the form (28.4), which are independent of the angle 
<f>, the system of equations under consideration has also solutions for which 
vi contains a factor e' m *, m being an integer. We are, however, interested 
only in the solution which corresponds to the first appearance of instability. 
The solutions with m # have never been studied in this respect. It is 
nevertheless natural to suppose that instability occurs first of all with respect 
to perturbations with m = 0, a supposition which is entirely confirmed by 
experimental results. 

It should also be borne in mind that, even for a given k, the solution of the 
form (28.4) is not unique. In general, a number of solutions with different 
values of co correspond to a given k. Again we are interested only in the one 
which gives the smallest value of R C r- 

It is found that a purely imaginary function co(k) corresponds to the solu- 
tion which gives the smallest R cr . Hence, when k = k cx , not only im co 
but co itself is zero. This means that the first instability of the steady flow 
between rotating cylinders leads to the appearance of another flow which is 
also steady. 



t For R slightly greater than R cr there is not one value of k, but a whole range, for which im a> > 0. 
However, it should not be thought that the resulting flow will be a superposition of flows with various 
periodicities. In reality, for each R a flow of definite periodicity occurs which stabilises the total 
flow. This periodicity, however, cannot be determined from the linearised equations (26.1). 



110 



Turbulence 



§28 



On account of the great complexity of the calculation,! numerical results 
have been obtained only for the case where the space between the cylinders 
is narrow {R 2 -R\ <^ R2). Fig. 13 shows an example of the curve separating 
the regions of unstable (shaded) and stable flow. The right-hand branch of 
the curve, corresponding to rotation of the two cylinders in the same direc- 
tion, is asymptotic to the line Q 2 #2 2 = Hii?i 2 . When the Reynolds number 
increases, for a given type of flow, the two numbers Cltf/jv and Q 2 #2 2 /v 
increase by equal factors. In Fig. 13 this corresponds to a movement upwards 
along a line through the origin having a given slope. In the right-hand part 
of the diagram (Di and Q 2 both positive), such lines for which a 2 R2 2 l&iRi 2 > 1 
do not meet the curve which bounds the region of instability. If, on the other 
hand, Q 2 .R 2 2 /fti.Ri 2 < 1, then for sufficiently large Reynolds numbers we 
enter the region of instability, in accordance with the condition (28.3). 




Fig. 13 



In the left-hand part of the diagram (Qi and Q 2 of opposite signs), any line 
through the origin eventually meets the curve, i.e. the flow can become un- 
stable for any value of the ratio £l 2 R2 2 l&iRi 2 , again in agreement with the 
results obtained above. For Q 2 = (when only the inner cylinder rotates), 
instability sets in when 

Oi = A\-Zvlhy/{hR 2 \ (28.5) 

where h = R 2 -Ri. 

The stability of the flow in the unshaded part of Fig. 13 does not mean, 
however, that the flow actually remains steady no matter how large R be- 
comes. Experiment shows that there is a limit beyond which stable non- 
steady flow becomes possible. In this region the steady flow is "metastable" : 
it is stable with respect to small perturbations, but unstable with respect to 
larger perturbations. If, owing to such perturbations, non-steady flow occurs 
in some region along the cylinders, it will subsequently "displace" the laminar 
flow in all space. This non-steady flow has, as soon as it appears, a large 
number of "degrees of freedom" (in the sense explained in §27), i.e. it is 
fully developed turbulence. 



t Further details may be found in the book by C. C. LtN, The Theory of Hydrodynamic Stability, 
Cambridge 1955. 



§29 Stability of flow in a pipe 111 

In the shaded part of Fig. 13, the flow again becomes turbulent for 
sufficiently large R, but there are, it seems, very few data concerning the 
way in which it appears. 

A limiting case of the flow between rotating cylinders, corresponding to 
large radii and small h = R2-R1, is flow between two parallel planes in 
relative motion (see §17). This flow is stable with respect to infinitely 
small perturbations for any value of R = Uhjv, where U is the relative 
velocity of the planes. Stable turbulent motion becomes possible, however, 
for values of R greater than about 1500. 

§29. Stability of flow in a pipe 

The steady flow in a pipe discussed in §17 loses its stability in an unusual 
manner. Since the flow is uniform in the ^-direction (along the pipe), the 
unperturbed velocity distribution vo is independent of x. Similarly to the 
procedure in §28, we can therefore seek solutions of equations (26.1) in the 
form 

Vl = ««*»-*> f[y,*). (29.1) 

Here also there is a value R = R cr for which yi = im w first becomes zero 
for some value of k. It is of importance, however, that the real part of the 
function co(k) is not now zero. 



R>R £ 




R=R C 
R<R« 



Fig. 14 



For values of R only slightly exceeding R cr , the range of values of k for 
which yi(k) > is small and lies near the point for which yi(k) is a maximum, 
i.e. dyi/dk = (as seen from Fig. 14). Let a slight perturbation occur in 
some part of the flow; it is a wave packet obtained by superposing a series of 
components of the form (29.1). In the course of time, the components for 
which y\(k) > will be amplified, while the remainder will be damped. 
The amplified wave packet thus formed will also be carried downstream with 
a velocity equal to the group velocity dcojdk of the packet; since we are now 
considering waves whose wave numbers lie in a small range near the point 
where dyijdk = 0, the quantity dco/dk « dcoi[dk is real, and is therefore the 
actual velocity of propagation of the packet. 



112 Turbulence §29 

This downstream displacement of the perturbations is very important, and 
causes the loss of stability to be totally different from that described in §28. 

We have seen that, for flow between rotating cylinders with R > R cr 
(when there are frequencies with im o> > 0), the original steady flow is no 
longer possible, since even small perturbations are increased to a finite 
amplitude. For flow in a pipe, however, the amplification of the perturbation 
is accompanied by its displacement downstream ; if we consider the flow at a 
given point in the pipe, it is found that the perturbation there is not amplified, 
but damped. It must also be borne in mind that, since in reality we have 
pipes of finite length, however great, any perturbation may be carried out of 
the pipe before it disrupts the laminar flow. Thus, even for R > R cr , steady 
flow in a pipe is effectively stable with respect to small perturbations, and can 
in principle take place for values of R considerably exceeding R cr . 

Since the perturbations increase with the co-ordinate x (downstream), 
and not with time at a given point, it is reasonable to investigate this type of 
instability as follows. Let us suppose that, at a given point, a continuously 
acting perturbation with a given frequency co is applied to the flow, and 
examine what will happen to this perturbation as it is carried downstream. 
Inverting the function o> = a)(k), we find what wave number k corresponds 
to the given (real) frequency co. If im k < 0, the factor e ikx increases with 
x, i.e. the perturbation is amplified downstream. The curve in the tuR-plane 
given by the equation im k(o), R) = defines the region of stability, and 
separates, for each R, the frequencies of perturbations which are amplified 
and damped downstream. 

The actual calculations are extremely complicated. A complete investi- 
gation has been made only for flow between two parallel planes (C. C. Lin, 
1946).f However, it is reasonable to suppose that the results will be quali- 
tatively the same for flow in a circular pipe. 

The limiting curve for flow between two planes is schematically shown in 
Fig. 15. The shaded area within the curve is the region of instability. As 
R -> oo, both branches of the curve are asymptotic to the R-axis.J For 
the smallest value of R at which undamped perturbations are possible we 
find by calculations R cr « 7700, R being defined as Uhjv, with h the distance 
between the planes and U the fluid velocity averaged across this distance. 

Thus, for any frequency between zero and a certain maximum value, there 
is a finite range of R values for which perturbations with the frequency con- 
cerned will be amplified. It is interesting to note that a small but finite 
viscosity of the fluid has, in a sense, a destabilising effect in comparison with 
the situation for a strictly ideal fluid. For, when R -> oo, perturbations with 
any finite frequency are damped, but when a finite viscosity is introduced we 
eventually reach a region of instability; a further increase in the viscosity 
(decrease in R) finally brings us out of this region. 



f A detailed account is given by C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge 1955. 
{ The asymptotic equations of the two branches for large R are a>A/C7 = 50/R a/l1 , coh/U = 11-2/R 3 / 7 . 



§29 



Stability of flow in a pipe 



113 



These calculations, however, do not answer the question whether, for 
sufficiently large R, flow in a pipe does not also exhibit true instability with 
respect to infinitely small perturbations, i.e. instability resulting in the 
amplification of perturbations with time at a given point. We shall outline 
the mathematical significance of such an instability. Let us consider some 
small perturbation which occurs at time t = in a finite region. Expanding 
it as a Fourier integral with respect to x, we can write it as 



l\M)e ik{x -^dk> 



where f(x) is a function describing the initial perturbation. In the course of 
time, each Fourier component of the perturbation will vary as e ia>t , with a 
frequency u> = a>{k, R), so that the whole perturbation at time t will be given 
by the integral 

\(f(tj)e mx -®- i0)t d{;dk. 

Since f(x) is zero except in a finite region, x- | has a finite range of values. 
Hence the behaviour of the integral for large t is essentially determined by 
the behaviour of the integral 

f e -iaUc)tdk. 
If this integral tends to infinity with t f the flow is in fact absolutely unstable. 




Fig. IS 

No such investigation has yet been made, even for a particular case. 
However, the experimental results concerning flow in pipes give reason to 
suppose that there is no true instability with respect to arbitrarily small 
perturbations for any R. This is indicated by the fact that, the more care- 
fully perturbations at the entrance to the pipe are prevented, the larger the 
Reynolds numbers for which laminar flow can be observed.f 



t Laminar flow has actually been observed up to R« 50,000, where R = Ud[v, d being the diameter 
of the pipe and U the mean velocity over its cross-section. 



114 Turbulence §30 

However, the experimental results also show that there is another critical 
Reynolds number (which we denote by R c /); this determines the limit 
beyond which stable non-steady flow can exist (cf. the end of §27). If, in 
any section of the pipe, turbulent flow occurs, then for R < R cr ' the turbulent 
region will be carried downstream and will diminish in size until it disappears 
altogether; if, on the other hand, R > R cr ', the turbulent region will extend 
in the course of time to include more and more of the flow. If perturbations 
of the flow occur continually at the entrance to the pipe, then for R < R cr ' 
they will be damped out at some distance down the pipe, no matter how strong 
they are initially. If, on the other hand, R > R cr ', the flow becomes turbulent 
throughout the pipe, and this can be achieved by perturbations which are the 
weaker, the greater R.f Thus laminar flow in a pipe with R > R cr ' is 
metastable, being unstable with respect to perturbations of finite intensity; 
the necessary intensity is the smaller, the greater R. 

As has been mentioned at the end of §28, non-steady flow arising by the 
disruption of metastable laminar flow is already fully-developed turbulence. 
In this sense the appearance of turbulence in a pipe is essentially different 
from the appearance of turbulence owing to the absolute instability of steady 
flow past finite bodies. In the latter case non-steady flow seems to appear 
in a continuous manner as we pass through R cr , the number of degrees of 
freedom increasing gradually (as explained in §§26 and 27). For flow in a 
pipe, however, turbulence appears discontinuously. This difference causes, 
in particular, the different dependence of the drag on the Reynolds number 
in the two cases. For example, if we consider the motion of any body in a 
fluid, the drag force F on it is continuous at R = R cr , where steady flow 
becomes absolutely unstable. At this point the curve F = F(R) can have 
only a bend corresponding to the change in the nature of the flow. For 
flow in a pipe, on the other hand, there are essentially two different laws of 
drag for R > R cr : one for steady flow, and the other for turbulent flow. 
The drag is discontinuous for whatever value of R marks the transition from 
one type of flow to the other. 

§30. Instability of tangential discontinuities 

Flows in which two layers of incompressible fluid move relative to each 
other, one "sliding" on the other, are absolutely unstable if the fluid is ideal; 
the surface of separation between these two fluid layers would be a surface of 
tangential discontinuity, on which the fluid velocity tangential to the surface 
is discontinuous. We shall see below (§35) what is the actual nature of the 
flow resulting from this instability; here we shall prove the above statement. 

If we consider a small portion of the surface of discontinuity and the flow 
near it, we may regard this portion as plane, and the fluid velocities vi and 
V2 on each side of it as constants. Without loss of generality we can suppose 



t For a pipe of circular cross-section R cr ' lies between 1600 and 1700. For flow between parallel 
planes, turbulent flow has been observed from R = 1400 upwards. 



§30 Instability of tangential discontinuities 115 

that one of these velocities is zero ; this can always be achieved by a suitable 
choice of the co-ordinate system. Let V2 = 0, and vi be denoted by v 
simply ; we take the direction of v as the ff-axis, and the s-axis along the normal 
to the surface. 

Let the surface of discontinuity receive a slight perturbation, in which all 
quantities — the co-ordinates of points on the surface, the pressure, and the 
fluid velocity — are periodic functions, proportional to e i{kx ~° >t) . We consider 
the fluid on the side where its velocity is v, and denote by v' the small change 
in the velocity due to the perturbation. According to the equations (26.1) 
(with constant vo = v and v — 0), we have the following system of equations 
for the perturbation v' : 

dv' grad^)' 

diw' = 0, + (v.grad)v' = . 

8t p 

Since v is along the #-axis, the second equation can be rewritten 

dv' eV gradp' 

f- v = . (JU.l) 

dt dx p 

If we take the divergence of both sides, then the left-hand side gives zero by 
virtue of diw' = 0, so that p' must satisfy Laplace's equation: 

AP' = 0. (30.2) 

Let £ = Z,(x> t) be the displacement in the sr-direction of points on the 
surface of discontinuity, due to the perturbation. The derivative dt,\dt 
is the rate of change of the surface co-ordinate £ for a given value of x. 
Since the fluid velocity component normal to the surface of discontinuity 
is equal to the rate of displacement of the surface itself, we have to the 
necessary approximation 

dl\dt = v'z-vdt/dx, (30.3) 

where, of course, the value of v' z on the surface must be taken. 

We seek p' in the form p' = f(z) *«**-«*>. Substituting in (30.2), we 
have for f(z) the equation d 2 //ds 2 - k 2 f = 0, whence / = constant xe** 2 . 
Suppose that the space on the side under consideration (side 1) corresponds to 
positive values of z. Then we must take/ = constant xr* z , so that 

p' = constant x «***-**> <r* z . (30.4) 

Substituting this expression in the ^-component of equation (30.1), we findf 

v'z = kp'xjip^kv-oi). (30.5) 

The displacement £ may also be sought in a form proportional to the same 
exponential factor e t(Jex ~ cot \ and we obtain from (30.3) v' z = i£(kv — a). 



t The case kv = co, though possible in principle, is not of interest here, since instability can arise 
only from complex frequencies to, not from real a). 



116 Turbulence §31 

This gives, instead of (30.5), 

P'l = -t P i{kv-a>flk. (30.6) 

The pressure p\ on the other side of the surface is given by a similar formula, 
where now v = and the sign is changed (since in this region z < 0, and all 
quantities must be proportional to e kz , not e~ kz ). Thus 

p' 2 = faafijk. (30.7) 

We have written different densities p\ and p2 in order to include the case 
where we have a boundary separating two different immiscible fluids. 

Finally, from the condition that the pressures p\ and p\ are equal on 
the surface of discontinuity, we obtain pi(kv — co) 2 = — p 2 a) 2 , from which 
the desired relation between co and k is found to be 

Pi± W(pm) , on ox 

o) = kv . (30.8) 

P1+P2 

We see that o> is complex, and there are always co having a positive imagi- 
nary part. Thus tangential discontinuities are unstable, even with respect 
to infinitely small perturbations. In this form the result is true for very 
small viscosities, i.e. for very large R. In this case it is meaningless to 
distinguish instability of the type that is "carried along" from true absolute 
instability, since, as k increases, the imaginary part of o> increases without 
limit, and hence the "amplification coefficient" of the perturbation as it is 
carried along may be as large as we please. 

When finite viscosity is taken into account, the tangential discontinuity is 
no longer sharp; the velocity changes from one value to another across a 
layer of finite thickness. The problem of the stability of such a flow is 
mathematically entirely similar to that of the stability of flow in a laminar 
boundary layer with a point of inflexion in the velocity profile (§41). The 
experimental results indicate that instability sets in very soon. 

§31. Fully developed turbulence 

Turbulent flow at fairly large Reynolds numbers is characterised by the 
presence of an extremely irregular variation of the velocity with time at each 
point. This is called fully developed turbulence. The velocity continually 
fluctuates about some mean value, and it should be noted that the amplitude 
of this variation is in general not small in comparison with the magnitude of 
the velocity itself. A similar irregular variation of the velocity exists between 
points in the flow at a given instant. The paths of the fluid particles in turbu- 
lent flow are extremely complicated, resulting in an extensive mixing of the 
fluid. 

As has been mentioned in the previous section, turbulent flow has a very 
large number of degrees of freedom. The values of the initial phases fa 
corresponding to these degrees of freedom are determined by the initial 



§31 Fully developed turbulence 117 

conditions of the flow. The specification of the exact initial conditions 
which would determine the value of so many quantities is, however, so un- 
realistic that even to put the problem in this form is physically meaningless. 

The position here is similar to what would happen if we attempted to 
consider the motion of all the molecules forming a macroscopic body, using 
the equations of mechanics; here again the problem of specifying the initial 
conditions which determine the initial values of the co-ordinates and velocities 
of all the molecules, and then integrating the equations of motion, is physically 
meaningless. The analogy extends further. A macroscopic body, regarded 
as composed of individual molecules, has an enormous number of degrees 
of freedom. An exact microscopic description of the state of the body would 
involve a determination of the co-ordinates and velocity of every particle 
composing it. The exact manner in which these quantities vary with time 
depends on their values at the initial instant. However, owing to the extreme 
complexity and irregularity of the motion of the molecules, we may suppose 
that, over a sufficiently long interval of time, the velocities and co-ordinates of 
the molecules take all possible sets of values, so that the effect of the initial 
conditions is smoothed out and disappears. This, as is well known, makes 
possible a statistical discussion of macroscopic bodies. 

A similar situation occurs in turbulent flow. For an exact description of 
the time variation of the velocity distribution in the moving fluid, the values 
of all the initial phases /?/ would have to be given; the values of all the phases 
<f>j = ajjt + fa at every instant would then be known. We have seen that, what- 
ever the initial phases /fy, over a sufficiently long interval of time the fluid 
passes through states arbitrarily close to any given state, defined by any 
possible choice of simultaneous values of the phases <f>j. Hence it follows that, 
in the consideration of turbulent flow, the actual initial conditions cease to 
have any effect after sufficiently long intervals of time. This shows that the 
theory of turbulent flow must be a statistical theory. No complete quantitative 
theory of turbulence has yet been evolved. Nevertheless, several very impor- 
tant qualitative results are known, and the following sections give an account 
of these. 

We introduce the concept of the mean velocity, obtained by averaging over 
long intervals of time the actual velocity at each point. By such an averaging 
the irregular variation of the velocity is smoothed out, and the mean velocity 
varies smoothly from point to point. In what follows we shall denote the mean 
velocity by u = v. The difference v' = v— u between the true velocity 
and the mean velocity varies irregularly in the manner characteristic of tur- 
bulence ; we shall call it the fluctuating part of the velocity. 

Let us consider in more detail the nature of this irregular motion which is 
superposed on the mean flow. This motion may in turn be qualitatively 
regarded as the superposition of turbulent eddies of different sizes ; by the size 
of an eddy we mean the order of magnitude of the distances over which the 
velocity varies appreciably. As the Reynolds number increases, large eddies 
appear first; the smaller the eddies, the later they appear. For very large 



118 Turbulence §31 

Reynolds numbers, eddies of every size from the largest to the smallest are 
present. An important part in any turbulent flow is played by the largest 
eddies, whose size is of the order of the dimensions of the region in which the 
flow takes place ; in what follows we shall denote by / this order of magnitude 
for any given turbulent flow. These large eddies have the largest amplitudes. 
The velocity in them is comparable with the variation of the mean velocity 
over the distance /; we shall denote by Am the order of magnitude of this 
variation.-)- The frequencies corresponding to these eddies are of the order 
of ujl, the ratio of the mean velocity u (and not its variation Am) to the dimen- 
sion /. For the frequency determines the period with which the flow pattern 
is repeated when observed in some fixed frame of reference. Relative to such 
a system, however, the whole pattern moves with the fluid at a velocity of the 
order of w. 

The small eddies, on the other hand, which correspond to large frequencies, 
participate in the turbulent flow with much smaller amplitudes. They may 
be regarded as a fine detailed structure superposed on the fundamental large 
turbulent eddies. Only a comparatively small part of the total kinetic energy 
of the fluid resides in the small eddies. 

From the picture of turbulent flow given above, we can draw a conclusion 
regarding the manner of variation of the fluctuating velocity from point to 
point at any given instant. Over large distances (comparable with /), the 
variation of the fluctuating velocity is given by the variation in the velocity 
of the large eddies, and is therefore comparable with Am. Over small 
distances (compared with /), it is determined by the small eddies, and is 
therefore small (compared with Am)4 The same kind of picture is obtained if 
we observe the variation of the velocity with time at any given point. Over 
short time intervals (compared with T ~ l/u), the velocity does not vary 
appreciably; over long intervals, it varies by a quantity of the order of Am. 

The length / appears as a characteristic dimension in the Reynolds number 
R, which determines the properties of a given flow. Besides this Reynolds 
number, we can introduce the qualitative concept of the Reynolds numbers 
for turbulent eddies of various sizes. If A is the order of magnitude of the 
size of a given eddy, and v\ the order of magnitude of its velocity, then 
the corresponding Reynolds number is defined as R^ ~ v\^l v - This number 
is the smaller, the smaller the size of the eddy. 

For large Reynolds numbers R, the Reynolds numbers R\ of the large 
eddies are also large. Large Reynolds numbers, however, are equivalent to 
small viscosities. We therefore conclude that, for the large eddies which are 
the basis of any turbulent flow, the viscosity is unimportant and may be 



t We are speaking here of the order of magnitude, not of the mean velocity itself, but of its variation 
(over distances of the order of /)> since it is this variation Au which characterises the velocity of the 
turbulent flow. The mean velocity itself can have any magnitude, depending on the frame of reference 
used. 

It may also be mentioned that experimental results indicate that the size of the largest eddies is 
actually somewhat less than I, and their velocity is somewhat less than Am. 

J But large compared with the variation of the mean velocity over these small distances. 



§31 Fully developed turbulence 119 

equated to zero, so that the motion of these eddies obeys Euler's equation. In 
particular, it follows from this that there is no appreciable dissipation of 
energy in the large eddies. 

The viscosity of the fluid becomes important only for the smallest eddies, 
whose Reynolds number is comparable with unity. We denote the size of 
these eddies by A , which we shall determine in the next section. It is in 
these small eddies, which are unimportant as regards the general pattern of 
a turbulent flow, that the dissipation of energy occurs. 

We thus arrive at the following conception of energy dissipation in turbu- 
lent flow. The energy passes from the large eddies to smaller ones, practi- 
cally no dissipation occurring in this process. We may say that there is 
a continuous flow of energy from large to small eddies, i.e. from small to 
large frequencies. This flow of energy is dissipated in the smallest eddies, 
where the kinetic energy is transformed into heat.f 

Since the viscosity of the fluid is important only for the smallest eddies, 
we may say that none of the quantities pertaining to eddies of sizes A > Ao 
can depend on v (more exactly, these quantities cannot be changed if v 
varies but the other conditions of the motion are unchanged). This circum- 
stance reduces the number of quantities which determine the properties of 
turbulent flow, and the result is that similarity arguments, involving the dimen- 
sions of the available quantities, become very important in the investigation 
of turbulence. 

Let us apply these arguments to determine the order of magnitude of the 
energy dissipation in turbulent flow. Let e be the mean dissipation of 
energy per unit time per unit mass of fluid.} We have seen that this 
energy is derived from the large eddies, whence it is gradually transferred to 
smaller eddies until it is dissipated in eddies of size ~ Ao. Hence, although 
the dissipation is ultimately due to the viscosity, the order of magnitude of e 
can be determined only by those quantities which characterise the large 
eddies. These are the fluid density p, the dimension / and the velocity 
Am. From these three quantities we can form only one having the dimensions 
of e, namely erg/g sec = cm 2 /sec 3 . Thus we find 

e ~ (Am)3//, (31.1) 

and this determines the order of magnitude of the energy dissipation in turbu- 
lent flow. 

In some respects a fluid in turbulent motion may be qualitatively described 
as having a "turbulent viscosity" j/ turb which differs from the true kinematic 
viscosity v. Since v turb characterises the properties of the turbulent flow, 
its order of magnitude must be determined by p, Aw and /. The only quantity 
that can be formed from these and has the dimensions of kinematic viscosity 



t For a steady state to be maintained, it is of course necessary that external energy sources should 
be present which continually supply energy to the large eddies. 

X In this chapter e denotes the mean dissipation of energy, and not the internal energy of the 
fluid. 



120 Turbulence §32 

is /Am, and therefore 

vturb ~ /Aw. (31.2) 

The ratio of the turbulent viscosity to the ordinary viscosity is consequently 
v tuTbl v ~ R> *- e « i* ^creases with the Reynolds number. f 
The energy dissipation e is expressed in terms of v turb by 

e - vturb(A M //)2 (31.3) 

in accordance with the usual definition of viscosity. Whereas v determines 
the energy dissipation in terms of the space derivatives of the true velocity, 
v turb re l a tes it to the gradient ( ~ Aw//) of the mean velocity. 

We may also apply similarity arguments to determine the order of mag- 
nitude Ap of the variation of pressure over the region of turbulent flow. 
The only quantity having the dimensions of pressure which can be formed 
from p, I and Aw is p(Au) 2 . Hence we must have 

Ap ~ P (Au)K (31.4) 

§32. Local turbulence 

Let us now consider the properties of the turbulence as regards eddy sizes 
A which are small compared with the fundamental eddy size /. We shall 
refer to these properties as local properties of the turbulence. We shall 
consider fluid that is far from all solid surfaces (more precisely, that is at 
distances from them large compared with A). 

It is natural to assume that such small-scale turbulence, far from solid 
bodies, is isotropic. This means that, over regions whose dimensions are 
small compared with /, the properties of the turbulent flow are independent 
of direction ; in particular, they do not depend on the direction of the mean 
velocity. It must be emphasised that here, and everywhere in the present 
section, when we speak of the properties of the turbulent flow in a small region 
of the fluid, we mean the relative motion of the fluid particles in that region, 
and not the absolute motion of the region as a whole, which is due to the larger 
eddies. 

It is found that several very important results concerning the local pro- 
perties of turbulence can be obtained immediately from similarity arguments. 
These results are due to A. N. Kolmogorov and to A. M. Obukhov (1941). 
To obtain them, we shall first determine which parameters can be involved in 
the properties of turbulent flow over regions small compared with / but large 
compared with the distances Xq at which the viscosity of the fluid begins to be 
important. It is these intermediate distances which we shall discuss below. The 



f In reality, however, a fairly large numerical coefficient should be included. This is because, as 
mentioned above, / and Am may differ quite considerably from the actual scale and velocity of the 
turbulent flow. The ratio nurb/" may be more accurately written vturb/y ~ R/R<sr> which formula 
takes into account the fact that vturb and v must in reality be comparable in magnitude not for R ~ 1, 
but for R ~ R cr . 



§32 Local turbulence 121 

parameters in question are the fluid density p and another quantity charac- 
terising any turbulent flow, the energy e dissipated per unit time per unit mass 
of fluid. We have seen that e is the "energy flux" which continually passes 
from larger to smaller eddies. Hence, although the energy dissipation is 
ultimately due to the viscosity of the fluid and occurs in the smallest eddies, 
the quantity € is determined by the properties of larger eddies. It is natural 
to suppose that (for given p and e) the local properties of the turbulence are 
independent of the dimension / and velocity Am of the flow as a whole. The 
fluid viscosity v also cannot appear in any of the quantities in which we are 
at present interested (we recall that we are concerned with distances A > Ao). 
Let us determine the order of magnitude v\ of the turbulent velocity varia- 
tion over distances of the order of A. It must be determined only by p, e 
and, of course, the distance A itself. From these three quantities we can 
form only one having the dimensions of velocity, namely (eA)*. Hence we 
can say that the relation 

v x ~ (eA)* (32.1) 

must hold. We thus reach a very important result : the velocity variation over 
a small distance is proportional to the cube root of the distance {Kolmogorov 
and Obukhov's law). The quantity v\ may also be regarded as the velocity 
of turbulent eddies whose size is of the order of A.f 

Let us now put the problem somewhat differently, and determine the order 
of magnitude v 7 of the velocity variation at a given point over a time interval 
t which is short compared with the time T ~ lju characterising the flow as 
a whole. To do this, we notice that, since there is a net mean flow, any given 
portion of the fluid is displaced, during the interval t, over a distance of 
the order of ru, u being the mean velocity. Hence the portion of fluid which 
is at a given point at time t will have been at a distance ru from that point 
at the initial instant. We can therefore obtain the required quantity v r by 
direct substitution of ru for A in (32.1): 

v T ~ (era)*. (32.2) 

Thus the velocity variation over a time interval t is proportional to the cube 
root of the interval. 



t The variation v x of the velocity over small distances is fundamentally the variation in the fluc- 
tuating part of the velocity; the variation of the mean velocity over small distances is small compared 
with the variation of the fluctuating velocity over those distances. 

The relation (32.1) may be obtained in another way by expressing a constant quantity, the dis- 
sipation e, in terms of quantities characterising the eddies of size A; e must be proportional to the 
squared gradient of the velocity v x and to the appropriate turbulent viscosity coefficient 

fturb.A ~ C A^ 
(cf. (31.2), (31.3)): 

€ ~ "turb,A(WA) 2 ~ *> A 3 /A, 
whence we obtain (32.1). 



1 22 Turbulence §32 

The quantity v T must be distinguished from v r \ the variation in velocity 
of a portion of fluid as it moves about. This variation can evidently depend 
only on p and e, which determine the local properties of the turbulence, and 
of course on t itself. Forming the only combination of p, e and t that has 
the dimensions of velocity, we obtain 

v T ' ~ (er)*. (32.3) 

Unlike the velocity variation at a given point, it is proportional to the square 
root of t, not to the cube root. It is easy to see that, for r small compared 
with T, v T ' is always less than v r .\ 
Using the expression (31.1) for e, we can rewrite (32.1) as 

v x ~ Am(A//)*. (32.4) 

Similarly, we can write v T as 

v T ~ A^r/T)*, (32.5) 

where T ~ Iju. 

Let us now find at what distances the fluid viscosity begins to be important. 
These distances Ao also determine the order of magnitude of the size of the 
smallest eddies in the turbulent flow (called the "internal scale" of the tur- 
bulence, in contradistinction to the "external scale" /). To determine A , 
we form the Reynolds number R A ~ v x \jv\ using (32.4), we obtain 

R A ~ Am-A4/3/ v /i/3. 

Introducing the Reynolds number R ~ /Am/i/ for the flow as a whole, we 
can rewrite this as R A ~ R(X/lf. The order of magnitude of A is that for 
which R A ~ 1. Hence we find 

Ao - //R*. (32.6) 

The same expression can be obtained by forming from />, e and v the only 
combination having the dimensions of length, namely Ao ~ (i^/e)*, and expres- 
sing e in terms of Am and / by means of (31.1). 

Thus the internal scale of the turbulence is inversely proportional to R f . 
For the corresponding velocity we have 

v Xa - Am/R*; (32.7) 

this also decreases when R increases. Finally, the order of magnitude of the 
frequencies corresponding to eddies of this size is too ~ w/Ao or 

coo ~ kR*/J. (32.8) 

This gives the order of magnitude of the upper end of the frequency spectrum 
of the turbulence; the lower end is at frequencies of the order of «//. Thus 
the frequency range increases with Reynolds number as R*. 



f The inequality v ' <^ v has in essence been assumed in the derivation of (32.2). 



§33 The velocity correlation 123 

Similar arguments enable us to determine the order of magnitude of the 
number of degrees of freedom of a turbulent flow. Let us denote by n the 
number of degrees of freedom per unit volume of the fluid ; n has the dimen- 
sions 1/cm 3 . This number can depend only on p, e and also the viscosity v, 
since the latter determines the lower limit of the sizes of the turbulent eddies. 
From these three quantities we can form only one having the dimensions 
1/cm 3 , namely (e/v 3 )*; this is just 1/Ao 3 , a result which might have been expec- 
ted. Thus we have 

n ~ 1/Ao 3 ~ R 9/4 // 3 . (32.9) 

The total number N of degrees of freedom is obtained by multiplying n 
by the volume of the region of turbulent flow, which is of the order of Z 3 :f 

N - R9/4. (32.10) 

Finally, let us consider the properties of the flow in regions whose dimen- 
sion A is small compared with Ao. In such regions the flow is regular and its 
velocity varies smoothly. Hence we can expand v x in a series of powers of 
A and, retaining only the first term, obtain v x = constant x A. The order of 
magnitude of the constant is v x /Ao, since for A ~ Ao we must have v x ~ v x • 
Substituting (32.6) and (32.7)' we find 

v x ~ Am-R*A//. (32.11) 

This formula may also be obtained directly by equating two expressions for 
the energy dissipation e: the expression (Am) 3 // (31.1), which determines e 
in terms of quantities characterising the large eddies, and the expression 
v(w A /A) 2 , which determines e in terms of the velocity gradient (~ v x [\) 
for the eddies in which the energy dissipation actually occurs. 

PROBLEM 

Two fluid particles are at a small distance \ (^> A ) apart. Determine the order of magnitude 
of the time t required for the particles to move apart to a distance ^2 (A x <^ Aj <^ /). 

Solution. If A ^> A , we have from dimensional considerations dX/dt <~" (cA)*. Integrating 
this and using the fact that Ag ^> X lt we find t ~ (V/e)*. 

§33. The velocity correlation 

Formula (32.1) determines qualitatively the correlation of velocities in 
local turbulence, i.e. the relation between the velocities at two neighbouring 
points. Let us now introduce quantities which will serve to characterise this 



t Formulae (32.6)-(32.10) determine how the corresponding quantities vary with the Reynolds 
number. Quantitatively, however, it must be borne in mind that a considerable numerical factor 
may actually appear in all these formulae. The number of degrees of freedom, for example, must be 
of the order of unity not for R <-w 1, but for R ^ R cr . Hence we must write the ratio R/R CT in place 
of R in (32.10): 

N ~ (R/R cr ) 9/4 . 



124 Turbulence §33 

correlation quantitatively. 4. These may be, for instance, the components of 
the tensor 



Boc = (»2i-«>u)(»2*-«>i*)» (33.1) 

where V2 and vi are the fluid velocities at two neighbouring points, and the 
bar denotes an average with respect to time.J The radius vector from point 1 
to point 2 will be denoted by r; we suppose its magnitude r small compared 
with / (but not necessarily large compared with the internal scale of turbulence 
Ao). 

Since local turbulence is isotropic, the tensor J5^ cannot depend on any 
direction in space. The only vector that can appear in the expression for Bat 
is the radius vector r. In other words, Bm can contain, apart from the absolute 
magnitude r of r, only the unit tensor 8^ and the unit vector n in the direction 
of r. The most general form of such a tensor of rank two is 

B ik = A(r)h ik + B{r)nin k . (33.2) 

We take the co-ordinate axes so that one of them is in the direction of n, 
denoting the velocity component along this axis by v r and the component 
perpendicular to n by vt. The component B rr is then the mean square 
relative velocity of two neighbouring fluid particles along the line joining 
them. Similarly, Btt is the mean square transverse velocity of one particle 
relative to the other, while B r t is the mean value of the product of these two 
velocity components. Since n r = 1, n t = 0, we have from (33.2) 

Brr = A + B, B tt = A, Brt=0 (33.3) 

Let us now derive a relation between B rr and Btt. To do so, we first 
notice that the velocity variation over small distances is mainly due to the 
small eddies. The properties of the local turbulence do not depend on the 
large eddies that are superposed on it. Hence, to calculate the tensor Buc, 
it suffices to take the particular case of completely isotropic and homogeneous 
turbulent flow, in which the mean fluid velocity is zero.f f Expanding the 
parentheses in (33.1), we have 



Bik = VuVi]e + V2iV2k — ViiV2k — VMV2i. 



t The results given in this section are due to T. von Karman and L. Howarth (1938) and to A. 
N. Kolmogorov (1941). Similar relations for the temperature fluctuations in a non-uniformly heated 
turbulent flow are given later (see §54, Problems 3 and 4). 

J If there were no correlation between the velocities at the points 1 and 2, the mean values of the 
products in (33.1) would reduce to products of the mean value of each factor separately, and would 
therefore be zero. 

ft Such a flow can be imagined as that of a fluid subjected to strong agitation and then left to itself. 
Of course, the flow will certainly decay with time. The averaging in formula (33.1) must then, strictly 
speaking, be taken not as an averaging over time but as one over all possible positions of the points 
1 and 2 (for a given distance r between them) at a given instant. 



§33 The velocity correlation 125 



Since the flow is completely homogeneous and isotropic, we have vnvik 
= V2tV2k, and vuV2k = vikV2t- Thus 



Boc = 2vuvijc-2vuV2k- (33.4) 

We differentiate this expression with respect to the co-ordinates of point 2: 



dBijcfdx2k = —2vudv2kl8x2k- 

By the equation of continuity, however, 8v2kl&X2k = 0, so that dBijddx2k = 0. 
Since But is a function only of the components xt = X2i — xu of the vector 
r, differentiation with respect to X2k is equivalent to differentiation with 
respect to Xk> Substituting (33.2), we have after a simple calculation 
A' + B' -\-2B\r = 0, the prime denoting differentiation with respect to r. 
Substituting (33.3), we can write this as B' ' rr + 2(B rr — Btt)jr = 0, whence 
we have finally the general relation between B rr and Btt: 

2rB tt = d(r*Brr)ldr. (33.5) 

At distances r large compared with Ao, the velocity difference is propor- 
tional to r*, according to (32.1). The components of the tensor Bue for such 
r are therefore proportional to r f . Substituting in (33.5) B rr = constant xr f , 
Btt = constant xr s , we obtain the simple relation 

Btt = m r . (33.6) 

For distances r small compared with Ao, the velocity difference is propor- 
tional to r, and therefore B rr and Btt are proportional to r 2 . Formula (33.5) 
then gives the relation 

Btt = 2B rr . (33.7) 

At these distances (r <t Ao), Btt and B rr can also be separately expressed in 
terms of the mean energy dissipation e. We write B rr = ar 2 , where a is. 
constant, and combine (33.2), (33.3), (33.4), obtaining 



»u»2* = viiOus- ar 2 8 ijc +%ar 2 nink. 
Differentiating this relation, we find 



dvu dv2i _ dvu di)2i 

dxu d%2i dx±i &X2i 

Since this holds for arbitrarily small r, we can put xu = X2t, whence 



foi\ ., - foi dvi 
I I = 15a, = 0. 

\ dxi / 8xi dxi 

According to the general formula (16.3), however, we have for the mean 



126 Turbulence §33 

energy dissipation 



n / dvt dvi \ 2 [7 dv t \2 dvi dvi 1 

e = \v[ + = v][ + = 15av, 

\ dxi dxi / L\ dxi 1 Bxi dxi J 

whence a = ej!5v. We therefore obtain the following relations giving B rr 
and B u in terms of the mean energy dissipation :f 

B t = rW 2 M Brr = tW 2 /v. (33.8) 

We may also discuss the triple correlation 



Buci = {v2i- vu)(vz k -v lk )(v 2 i-vu). (33.9) 

We shall again suppose that the flow is completely homogeneous and iso- 
tropic. Let us first consider the auxiliary tensor vuvi k V2i. This tensor is 
symmetrical in the suffixes i and k, and by virtue of the isotropy it must, 
like Btk, be expressible in terms of «j and S^. The most general form of such 
a tensor is 



«ii*>i*0» = C(r)8 ik ni+D(r)(8nn k +S k flii) + F(r)nin k ni. (33.10) 

Differentiating with respect to xzu we have by the equation of continuity 



dvzi 



—(vuV lk V2l) = *>li*>lfc- — = 0. 
0X21 OX2\ 



Substituting the expression for vuvi k V2i, we have after a simple calculation 
(here omitted) two equations: 

d[r*(3C+2D + F)]ldr = 0, 

C' + 2(C+D)/r = 0. 

Integration of the former gives 3C+2D + F = constant/r 2 . For r = the 
functions C, D and F must remain finite. We must therefore put the constant 
equal to zero, so that 3C+2D+F = 0. From the two equations thus ob- 
tained we find 

D = -(C+irC), F = rC'-C. (33.11) 

We now expand the parentheses in (33.9). It is easy to see that, by virtue of 



t It might be thought that a possibility exists in principle of obtaining a universal formula, appli- 
cable to any turbulent flow, which should give B rr and Btt for all distances r that are small compared 
with /. In fact, however, there can be no such formula, as we see from the following argument. The 
instantaneous value of (w 2t - — v u ) (v 2 k —vm) might in principle be expressed as a universal function 
of the energy dissipation e at the instant considered. When we average these expressions, however, 
an important part will be played by the law of variation of e over times of the order of the periods 
of the large eddies (of size <*~> I), and this law is different for different flows. The result of the averaging 
therefore cannot be universal. 



§33 The velocity correlation 127 



the isotropy of the flow, the mean values vuVikVu and V2iV2kV2i are zero. 
For all three velocities in these products are taken at the same point; the 
only tensor in terms of which the tensor ViVtfvi could be expressed is therefore 
hoc. It is, however, impossible to construct a symmetrical tensor of rank 
three from unit tensors. Such mean values as vuvi k V2i and t>2i^2fc^iz> on 
the other hand, are equal in magnitude and opposite in sign, since the vector 
ni in (33.10) changes sign when points 1 and 2 are interchanged. The 
result is 



Bikl = 2(vi i V 1 ] c V2l + V\iV21cVli + VziWcVll). 

Substituting (33.10) and (33.11), we have the expression 

B m =2(rC + C)(8 ik m + 8 u n k + 8 k im) + 6{rC -G)tynm- (33.12) 

Again taking one of the co-ordinate axes parallel to n, we obtain the com- 
ponents of the tensor Bm". B rrr = — 12C, B r tt = —2(C+rC), B rrt = But 
= 0. Hence we see that the relation 

6B m = d(rBrrr)ldr (33.13) 

holds between the non-zero components B r tt and B rrr . 

Finally, it is also possible to find a relation between the components of 
the tensors Bik and Bm. To do so, we calculate the derivative d(vuV2k)ldt, 
recalling that a completely homogeneous and isotropic flow necessarily 
decays with time. Expressing the derivatives dvu/dt and dvzkjdt by means 
of the Navier-Stokes equation, we obtain 



d d d 8 /PiV2k\ 

-~iVuV2k) = - (VliVuV2k) - (VuV2kV2l) ~ ~ I - 

at dxu 0x21 ox\% \ a I 



d I P2V\% \ 

- +vAl(ai#>2fc) + vA2(t>li«>2&). 

OX2k \ P ' 

In using the properties of homogeneity and isotropy, it must be borne in mind 
that the sign of r changes when the points 1 and 2 are interchanged, and 
therefore the sign of the (first) space derivatives must be changed. The first 
two terms are therefore equal, and so are the last two terms. The third and 
fourth terms are zero. For, by virtue of the isotropy, the mean value p\V2k 

must be of the form f(r)rik. The divergence d(piV2k)I^X2k = pi 8®2kldx2k is 
zero. But the only centrally symmetric vector whose divergence is every- 
where zero is a constant times (l[r 2 )nk. Such a vector would become infinite 
for r = 0, which is impossible. The constant must therefore be zero. 
Thus 



— (viiV 2 k) = - 2- — (vuviMk) + 2v/\iv u v2k- (33. 14) 

ot dxu 



128 Turbulence §34 

Here we must substitute, in accordance with the formulae derived above, 



VliV2k = ^2i^2& — i-Bflfc, 



(33.15) 
— Ts(rB rrr ' — Brr^fiitijcni. 



In the former expression we replace V2iV2k by iv 2 Sue, using the complete 
homogeneity and isotropy of the flow: 



»u*>2* = i« a 8tt-&B tt . (33.16) 

The time derivative of the kinetic energy per unit mass \v 2 is just the energy 
dissipation -e; hence d(%v 2 )ldt = -§e. A simple, though lengthy, calcula- 
tion gives the equation 



_2- 



1 dB rr 1 d^Brrr) v d [ dB^ 

2 dt ~ 6r* dr r* Hr 



Kr)- (33 - 17) 



Since r is supposed small, we can with sufficient accuracy put r = on the 
left-hand side, i.e. neglect 8B rr /dt in comparison with e. Multiplying the 
resulting equation by r 4 , integrating over r, and using the fact that the cor- 
relation functions vanish for r = 0, we obtain the following relation between 
Brr and Brrr' 

Brrr= -Ur + 6vdBrr/dr. (33.18) 

The relation (33.18), like (33.13), holds for r either greater or less than Ao. 
For r > Ao, the viscosity term is small, and we have simply 

Brrr = -fer. (33.19) 

If r <^ Ao, we can substitute the expression (33.8) for B rr in (33.18), obtaining 
Brrr = 0; this is because B rr r in this case must be of the third order in r, 
and so the first-order terms must cancel.f 

§34. The turbulent region and the phenomenon of separation 

Turbulent flow is in general rotational. However, the distribution of the 
vorticity o>( = curl v) in the fluid has certain peculiarities in turbulent flow 
(for very large R): in "steady" turbulent flow past bodies, the whole volume 
of the fluid can usually be divided into two separate regions. In one of these 
the flow is rotational, while in the other the vorticity is zero, and we have 



t The ratio | B rrr lB„\ must have constant values in the ranges /^> r ^> A and r <^ A . The ex- 
perimental results show that in fact this quantity is approximately constant for all r, being about 0-4. 



§34 The turbulent region and the phenomenon of separation 129 

potential flow. Thus the vorticity is non-zero only in a part of the fluid 
(though not in general only in a finite part). 

That such a limited region of rotational flow can exist is a consequence of 
the fact that turbulent flow may be regarded as the motion of an ideal fluid, 
satisfying Euler's equations.! We have seen (§8) that, for the motion of an 
ideal fluid, the law of conservation of circulation holds. In particular, if 
at any point on a streamline co = 0, then the same is true at every point 
on that streamline. Conversely, if at any point on a streamline co =£ 0, 
then co does not vanish anywhere on the streamline. Hence it is clear that 
the existence of limited regions of rotational and irrotational flow is compatible 
with the equations of motion if the region of rotational flow is such that 
the streamlines within it do not penetrate into the region outside it. Such a 
distribution of co will be stable, and the vorticity will remain zero beyond 
the surface of separation. 

One of the properties of the region of rotational turbulent flow is that the 
exchange of fluid between this region and the surrounding space can occur in 
only one direction. The fluid can enter this region from the region of potential 
flow, but can never leave it. 

We should emphasise that the arguments given here cannot, of course, be 
regarded as affording a rigorous proof of the statements made. However, 
the existence of limited regions of rotational turbulent flow seems to be 
confirmed by experiment. 

The flow is turbulent both in the rotational and in the irrotational region. 
The nature of the turbulence, however, is totally different in the two regions. 
To elucidate the reason for this difference, we may point out the following 
general property of potential flow, which obeys Laplace's equation A<£ = 0. 
Let us suppose that the flow is periodic in the ry-plane, so that <f> involves 
x and y through a factor of the form e iJc i x+ik 2y. Then 

320/0*2+ #ty/0y* = -(k 1 2 + k 2 2 )<f> = -&<(>, 

and, since the sum of the second derivatives must be zero, the second deriva- 
tive of cf> with respect to z must equal <f> multiplied by a positive coefficient: 
d 2 cf>l8z 2 =k 2 cf>. The dependence of <f> on z is then given by a damping factor 
of the form e~ kz for 2 > (the unlimited increase given by e kz is clearly 
impossible). Thus, if the potential flow is periodic in some plane, it must be 
damped in the direction perpendicular to that plane. Moreover, the greater 
k\ and &2 (i.e. the smaller the period of the flow in the ry-plane), the more 
rapidly the flow is damped along the sr-axis. All these arguments remain 
qualitatively valid in cases where the motion is not strictly periodic, but has 
only some periodic quality. 

From this the following result is immediately obtained. Outside the region 
of rotational flow, the turbulent eddies must be damped, and must be so 



f The applicability of these equations to turbulent flow ends at distances of the order of A . The 
sharp boundary between rotational and irrotational flow is therefore defined only to within such 
distances. 



130 Turbulence §35 

the more rapidly, the smaller their size. In other words, the small eddies do 
not penetrate very far into the region of potential flow. Consequently, only 
the largest eddies are important in this region; they are damped at distances 
of the order of the (transverse) dimension of the rotational region, which is 
just the external scale of turbulence in this case. At distances greater than 
this dimension there is practically no turbulence, and the flow may be re- 
garded as laminar. 

We have seen that the energy dissipation in turbulent flow occurs in the 
smallest eddies; the large eddies do not involve appreciable dissipation, 
which is why Euler's equation is applicable to them. From what has been said 
above, we reach the important result that the energy dissipation occurs mainly 
in the region of rotational turbulent flow, and hardly at all outside that region. 

Bearing in mind all these properties of the rotational and irrotational 
turbulent flow, we shall henceforward, for brevity, call the region of rotational 
turbulent flow simply the region of turbulent flow or the turbulent region. 
In the following sections we shall discuss the form of this region in various 
cases. 

The turbulent region must be bounded in some direction by part of the 
surface of the body past which the flow takes place. The line bounding this 
part of the surface is called the line of separation. From it begins the surface 
of separation between the turbulent fluid and the remainder. The formation 
of a turbulent region in flow past a body is called the phenomenon of separa- 
tion. 

The form of the turbulent region is determined by the properties of the 
flow in the main body of the fluid (i.e. not in the immediate neighbourhood 
of the surface). A complete theory of turbulence (which does not yet exist) 
would have to make it possible, in principle, to determine the form of this 
region by using the equations of motion for an ideal fluid, given the position 
of the line of separation on the surface of the body. The actual position 
of the line of separation, however, is determined by the properties of the flow 
in the immediate neighbourhood of the surface (known as the boundary 
layer), where the viscosity plays a vital part (see §40). 

§35. The turbulent jet 

The form of the turbulent region, and some other basic properties of it, 
can be established in certain cases by simple similarity arguments. These 
cases include, among others, various kinds of free turbulent jet in a space 
filled with fluid (L. Prandtl, 1925). 

As a first example, let us consider the turbulent region formed when a 
flow is "separated" at an angle formed by two infinite intersecting planes 
(shown in cross-section in Fig. 16). For laminar flow (Fig. 3, §10), the flow 
along one side of the angle (AO, say) would turn smoothly and flow along 
the other side away from the angle (OB). In turbulent flow, the pattern is 
totally different. 

The flow along one side of the angle now does not turn on reaching the 



§35 The turbulent jet 131 

vertex, but continues in its former direction. A flow appears along the 
other side in the direction BO. The two flows "mix" in the turbulent 
regionjf the boundaries of this region are shown, dashed, in cross-section 
in Fig. 16. The origin of this region can be seen as follows. Let us imagine 
a flow in which a uniform stream along AO continues in the same direction, 
occupying the whole space above the plane AO and its continuation into the 
fluid to the right, while the fluid below this plane is at rest. In other words, 
we have a surface of separation (the plane AO produced) between fluid moving 
with constant velocity and stationary fluid. Such a surface of discontinuity, 
however, is unstable, and cannot exist in practice (see §30). This instability 
leads to mixing and the formation of a turbulent region. The flow along 
BO arises because fluid must enter the turbulent region from below. 



777777777777777^^^ V^ * 




Let us determine the form of the turbulent region. We take the ar-axis 
in the direction shown in Fig. 16, the origin being at O. We denote by 
Yi and Y% the distances from the xz-plane to the upper and lower boundaries 
of the turbulent region, and require to determine Y± and Y^ as functions of x. 
This can easily be done from similarity considerations. Since the planes 
are infinite in all directions, there are no constant parameters at our disposal 
having the dimensions of length. Hence it follows that Fi, Yi can only be 
directly proportional to the distance x: 

Yi = a;tanai, Y% = #tana2. (35.1) 

The proportionality coefficients are simply numerical constants; we write 
them as tan <xi, tan a2, so that <xi and <X2 are the angles between the two 
boundaries of the turbulent region and the a?-axis. Thus the turbulent region 
is bounded by two planes intersecting along the vertex of the angle. 

The values of <xi, <X2 depend only on the size of the angle, and not, for 
example, on the velocity of the main stream. They cannot be calculated 



f We recall that, outside the turbulent region, there is irrotational flow which gradually becomes 
laminar as we move away from the boundaries of this region. 



132 Turbulence §35 

theoretically; the experimental results for flow round a right angle are 
ai = 5°, a 2 = 10°.f 

The velocities of the flows along the two sides of the angle are not the 
same; their ratio is a definite number, again depending only on the size of 
the angle. When the angle is not close to it, one of the velocities is considerably 
the greater, namely that of the main stream, which is in the same direction 
(AO) as the turbulent region. For example, in flow round a right angle, the 
velocity along the plane AO is thirty times that along BO. 

We may also mention that the difference between the fluid pressures on 
the two sides of the turbulent region is very small. For example, in flow round 
a right angle it is found that p± — p% = 0-003p£/i 2 , where JJ\ is the velocity 
of the main stream (along AO), p\ the pressure in that stream, and p2 the 
pressure in the stream along BO. 

In the limiting case of flow round an angle of 2xr, we have simply the 
edge of a plate with fluid moving along both sides. The angle ai + a2 of the 
turbulent region is zero, i.e. there is no turbulent region; the velocities of the 
flows along the two sides of the plate become equal. As the angle AOB 
increases, a point is reached when the plane BO forms the lower boundary of 
the turbulent region; the angle AOB is by then obtuse. As the angle increases 
further, the turbulent region continues to be bounded by the plane BO on 
one side. Here we have simply a separation, with the line of separation along 
the vertex of the angle. The angle of the turbulent region remains finite. 

As a second example, let us consider the problem of a turbulent jet of 
fluid issuing from the end of a narrow tube into an infinite space filled with 
the same fluid. The problem of laminar flow in such a "submerged jet" has 
been solved in §23. At distances (the only ones we shall consider) large 
compared with the dimensions of the mouth of the tube, the jet is axially 
symmetrical, whatever the actual shape of the opening. 

Let us determine the form of the turbulent region in the jet. We take 
the axis of the jet as the #-axis, and denote by R the radius of the turbulent 
region; we require to determine R as a function of x (which is measured 
from the end of the tube). As in the previous example, this function is easily 
determined directly from similarity considerations. At distances large 
compared with the dimensions of the mouth of the tube, the actual shape and 
size of the opening cannot affect the form of the jet. Hence we have at our 
disposal no characteristic parameters of the dimensions of length. It therefore 
follows as before that R must be proportional to x: 

R = x tan a, (35.2) 

where the numerical constant tan a is the same for all jets. Thus the turbulent 



f Here, and elsewhere, we speak of experimental data on the velocity distribution in a transverse 
cross-section of the turbulent jet, reduced by means of calculations (W. Tollmien 1926) based on 
the mixing-length theory (see the final note to the present section). This theory contains an arbitrary 
constant, whose value is chosen so as to obtain the best possible agreement with experiment. 



§35 The turbulent jet 133 

region is a cone; the experimental value of the angle 2a is 25 to 30 degrees 

(Fig. 17).f 

The (time average) velocity distribution in a cross-section of the jet has 
the following properties. The flow is principally along the jet. The longitudi- 
nal velocity component falls off rapidly away from the axis of the jet; it be- 
comes fwo ("0 being the velocity on the axis) at a distance of only 0-351? 
from the axis, and at the boundary of the turbulent region it is of the order of 
0-01 mo- The transverse velocity component is approximately uniform in order 
of magnitude over the cross-section of the turbulent region, and at the 
boundary of this region it is about -0-025 « , being there directed into the 
jet. This transverse component causes a flow into the turbulent region. The 
velocity distribution outside the turbulent region (for a given angle a) can 
be determined theoretically (see Problem 1). 






// 

Fig. 17 

The velocity in the jet also falls off as we move away from the mouth of 
the tube. The law of this decrease is easily found. To do so, we use the 
following method. The total flux of momentum through a spherical surface 
centred at the tube mouth must be independent of the radius of the surface. 
The momentum flux density in the jet is of the order of pu 2 , where u is 
of the order of some mean velocity in the jet; this is the only quantity of the 
right dimensions that can be formed from the fluid density p, the velocity u, 
and the distance x. The area of the part of the jet cross-section where u is 
appreciably different from zero is of the order of R 2 . Hence the total momen- 
tum flux is of the order of pu 2 R 2 . Equating this to a constant and putting 
R = constant xx, we obtain 

u ~ constant/*, (35.3) 

i.e. the velocity diminishes inversely as the distance from the mouth of the 
tube. 



t Some dependence of the constant a on the initial conditions (velocity profile) in the tube mouth 
is observed experimentally. It is reasonable to suppose that this dependence is due to the effect of 
the finite dimensions of the opening, an effect which would disappear at greater distances. 



134 Turbulence §35 

The amount Q of fluid which passes per unit time through a cross-section of 
the turbulent region of the jet is of the order of the product of its area ( ~ R 2 ) 
and the mean velocity u. Substituting, we findf 

Q = Bx. (35.4) 

Thus the discharge through a cross-section of the turbulent region in- 
creases with x, i.e. some fluid is, as it were, entrained in the turbulent 
region. $ The constant which appears in (35.4) may be determined as follows. 
At distances of the order of the dimensions of the tube mouth, Q must be- 
come the amount Q of fluid emitted from the tube per unit time, which is 
fixed for any particular jet. Hence we see that B ~ Qo/a, where a gives the 
transverse dimension of the tube mouth (e.g. the radius, if the opening is 
circular). Thus we can write 

B = cQola, (35.5) 

where c is a numerical constant which depends only on the form of the open- 
ing. If the latter is circular, c is found by experiment to be about 1 -5. 

The flow in any section of the length of the jet is characterised by the 
Reynolds number for that section, defined as uR[v. By virtue of (35.2) 
and (35.3), however, the product uR is constant along the jet, so that the 
Reynolds number is the same for all such sections. It can be taken, for 
instance, as Bjpv. The constant B which appears here is the only parameter 
which determines the flow in the jet. When the "strength" Q of the jet 
increases (the value of a remaining constant), the Reynolds number Bjpv 
eventually reaches a critical value, after which the flow simultaneously 
becomes turbulent along the whole length of the jet.ff 



t If two variable quantities which vary within wide limits are always of the same order of mag- 
nitude, then they must be proportional. Hence, in this case (and in similar cases), we can write 
precisely Q = constant X a; in place of Q — constant X x. 

% The total mass flux through any infinite plane across the jet is infinite, i.e. a jet issuing into an 
infinite space carries with it an infinite amount of fluid. 

ft In order to make more detailed calculations for various kinds of turbulent flow, it is customary 
to employ certain "semi-empirical" theories, based on assumptions concerning the dependence of the 
turbulent viscosity coefficient on the gradient of the mean velocity. For example, in Prandtl's theory 
it is assumed that (for plane flow) 

Vturb = l 2 \8uxldy\, 

where the dependence of / (called the mixing length) on the co-ordinates is chosen in accordance 
with the results of similarity arguments; for instance, in free turbulent jets we put I = ex, c being an 
empirical constant. Such theories usually give good agreement with experiment, and are therefore 
useful for interpolatory calculations. However, it is not possible to give universal values to the em- 
pirical constants which characterise each theory; for example, the value of the ratio of the mixing 
length I to the transverse dimension of the turbulent region has to be chosen differently in various 
particular cases. It should also be mentioned that good agreement with experimental results can be 
obtained with various expressions for the turbulent viscosity. 

A more detailed account of these theories is given by L. G. LoItsyanskiI, Aerodynamics of Boundary 
Layers (Aerodinamika pogranichnogo sloya), Moscow 1941; G. N. Abramovich, Free Turbulent Jets 
of Liquids and Gases (Turbulentnye svobodnye strui zhidkostei i gazov), Moscow 1948; H. Schuchting, 
Boundary Layer Theory, Pergamon Press, London 1955. 



§35 The turbulent jet 135 

PROBLEMS 

Problem 1. Determine the mean flow in the jet outside the turbulent region. 

Solution. We take spherical co-ordinates r, 6, $, with the polar axis along the axis of the 
jet, and the origin at its point of emergence. Because the jet is axially symmetrical, the 
component u^ of the mean velocity is zero, while ug and u r are functions only of r and 6. 
The same arguments as were used in the problem of the laminar jet (§23) show that ug and 
ur must be of the forms ug — f(0)/r, u r — F(0)Jr. Outside the turbulent region we have 
potential flow, i.e. curl u = 0, so that 8ur/d9 — 8(rue)/8r = 0. But rug is independent of r, 
so that BurJdO = (1/r) dF/dd = 0, whence F = constant = —b, say, or 

ur = -bjr. (1) 

From the equation of continuity, 



Id Id 

(r^ur) + —— —iug sin0) = 0, 
r 2 dr r sin 9 do 



we then obtain 



constant— b cos 6 
J sin0 

The constant of integration must be —b if the velocity is not infinite for d — it (it does not 
matter that / is infinite for = 0, since the solution in question refers only to the space 
outside the turbulent region, whereas B — lies inside that region). Thus 

6 (1 + cos 6) b 

u e = r-7— = - -cotJ0. (2) 

r sin 6 r 

The component of the velocity in the direction of the jet (u x ) and its absolute magnitude are 

b b cos 6 b 

u x = - = , u = . . (3) 

r x r sin$0 

The constant b can be related to the constant B in (35.4). Let us consider a segment of the 
cone formed by the turbulent region, bounded by two infinitely close cross-sections of the 
cone. The mass of fluid entering this segment per unit time is dO = —litrp sin a . ugdr 
= 2irbp(l +cos a)dr, while from formula (35.4) we have dQ = B dx = B cos a dr. Com- 
paring the two expressions, we obtain 

B cos a 
5 = . (4) 

27rp(l + cosa) 

At the boundary of the turbulent region, the velocity u is directed into this region, making 
an angle \(ir—a.) with the positive direction of the ar-axis. _ 

Let us compare the mean velocity u x inside the turbulent region (defined as u z = 
Qj-rrpR 2 = Bjirpx tan 2 a) with the velocity (uz) pot at the boundary of the region. Taking the 
first equation (3) with 9 = a, we find 

(u x ) P otlu x = 4(1 - cos a )' 

For a = 12°, this ratio is 0-011, i.e. the velocity at the boundary of the turbulent region is 
small compared with the mean velocity inside the region. 



136 Turbulence §36 

Problem 2. Determine the law of variation of size and velocity in a submerged turbulent 
jet issuing from an infinitely long thin slit. 

Solution. By the same reasoning as for the axial jet, we conclude that the turbulent 
region is bounded by two planes intersecting along the slit, i.e. the half-width of the jet is 
Y = x tan a. The momentum flux in the jet (per unit length of the slit) is of the order of 
pu 2 Y. The dependence of the mean velocity u on * is therefore given by u = constant/ V*- 
The discharge through a cross-section of the turbulent region is Q ~ pu Y, whence Q = con- 
stant X \/x. The experimental data give a value of 25° to 33° for the angle 2a of a plane- 
parallel jet (cf. the third footnote to this section). 

§36. The turbulent wake 

For Reynolds numbers considerably above the critical value, in flow past 
a solid body, a long region of turbulent flow is formed behind the body. This 
is called the turbulent wake. At distances large compared with the dimension 
of the body, simple arguments enable us to determine the form of this wake 
and the way in which the fluid velocity decreases there (L. Prandtl, 1926). 

As in the investigation of the laminar wake in §21, we denote by U the 
velocity of the incident stream, and take the direction of U as the #-axis. 
The fluid velocity at any point, averaged over the turbulent fluctuations, is 
written as U+u. Denoting by a some mean width of the wake, we shall 
find a as a function of x. If there is no lift, then at large distances from the 
body the wake is axially symmetrical and circular in cross-section ; in this case, 
a may be the radius of the wake. If a lift force is present, a direction is 
selected in the j#-plane, and the wake is not axially symmetrical at any distance 
from the body. 

The longitudinal fluid velocity component in the wake is of the order of 
U, while the transverse component is of the order of some mean value u 
of the turbulent velocity. The angle between the streamlines and the *-axis 
is therefore of the order of u\ U. The boundary of the wake is, as we know, the 
boundary beyond which the streamlines of the rotational turbulent motion 
cannot pass. Hence it follows that the angle between the boundary of the 
wake and the #-axis is also of the order of ujU. This means that we can write 

dajdx ~ uJU. (36.1) 

Next we use formulae (21.1), (21.2), which determine the forces on the 
body in terms of integrals of the fluid velocity in the wake (the velocity now 
being interpreted as its mean value). The region of integration in these 
integrals is of the order of a 2 . Hence an estimate of the integral gives 
F ~ pUua 2 , where F is of the order of the drag or the lift. Thus 

u ~ F/pUa*. (36.2) 

Substituting in (36.1), we find dajdx ~ F/pU 2 a 2 , from which we have by 
integration 

a ~ {Fx\pTJ 2 )K (36.3) 

Thus the width of the wake increases as the cube root of the distance from 



§37 Zhukovskii's theorem 137 

the body. For the velocity u, we have from (36.2) and (36.3) 

u ~ (FU/px*)*, (36.4) 

i.e. the mean fluid velocity in the wake is inversely proportional to xK 

The flow in any cross-section of the wake is characterised by the Reynolds 
number R ~ aujv. Substituting (36.2) and (36.3), we obtain 

R ~ FjvpUa ~ {pifPUxiPf. 

We see that this number is not constant along the wake, unlike what we found 
for the turbulent jet. At sufficiently large distances from the body, R becomes 
so small that the flow in the wake is no longer turbulent. Beyond this point 
we have the laminar wake, whose properties have been investigated in §21. 

In §21, Problem 2, formulae have been obtained which describe the flow 
outside the wake and far from the body. These formulae hold for flow 
outside the turbulent wake as well as outside the laminar wake. 

We may mention here some general properties of the velocity distribution 
round the body. Both inside and outside the turbulent wake, the velocity 
(by which we always mean u) decreases away from the body. However, the 
longitudinal velocity u x falls off more rapidly (~ \\x 2 ) outside the wake 
than inside it. Far from the body, therefore, we may suppose u x to be 
zero outside the wake. We may say that u x falls from some maximum value 
on the axis of the wake to zero at the boundary of the wake. The transverse 
components %, u z at the boundary are of the same order of magnitude as 
they are inside the wake, diminishing rapidly as we move away from the 
wake at a given distance from the body. 

§37. Zhukovskii's theorem 

The velocity distribution round a body, described at the end of the last 
section, does not hold for exceptional cases where the thickness of the wake 
formed behind the body is very small compared with its width. A wake 
of this kind is formed in flow past bodies whose thickness (in the ^-direction) 
is small compared with their width (in the ^-direction) ; the length (in the 
direction of flow, the ^-direction) may be of any magnitude. That is, we are 
considering flow past bodies whose cross-section transverse to the flow is 
very elongated. These bodies include, in particular, wings, i.e. bodies whose 
width, or span, is large in comparison with their other dimensions. 

It is clear that, in such a case, there is no reason why the velocity com- 
ponent u y perpendicular to the plane of the turbulent wake should fall off 
appreciably at distances of the order of the thickness of the wake. On the 
contrary, this component will now be of the same order of magnitude inside 
the wake and at considerable distances from it, of the order of the span. 
Here, of course, we assume that the lift is not zero, since otherwise the trans- 
verse velocity practically vanishes. 



138 Turbulence §37 

Let us consider the vertical lift force F y resulting from such a flow. 
According to formula (21.2), it it given by the integral 

F y = - P UJju y dydz, (37.1) 

where, on account of the nature of the distribution of u y , the integration 
must now be taken over the whole transverse plane. Furthermore, since the 
thickness of the wake (in the ^-direction) is small, while the velocity u y 
inside the wake is not large compared with its value outside, we can with 
sufficient accuracy take the integration over y to be over the region outside 
the wake, writing 



Vx 



J u y Ay « J u y dy+ \u y dy, 

—oo y t —oo 

where y 1 and y 2 are the co-ordinates of the boundaries of the wake (Fig. 18). 

-£ !/ 






\ 7/2 

\ /I 



Fig. 18 



Outside the wake, however, we have potential flow, and u y = d<f>/dy; 
bearing in mind that <f> = at infinity, we therefore obtain 

j Uydy = <f> 2 -<f>i, 

where $2 and $1 are the values of the potential on the two sides of the wake. 
We may say that <f>2-<f>i is the discontinuity of the potential at the surface of 
discontinuity which may be substituted for a thin wake. The derivative 
Uy = dtfajdy must remain continuous. A discontinuity in the velocity com- 
ponent normal to the surface of the wake would mean that some quantity of 
fluid flows into the wake; in the approximation in which the thickness of the 
wake is neglected, however, this inflow must be zero. Thus we replace the 
wake by a surface of tangential discontinuity. Next, in the same approxima- 
tion, the pressure also must be continuous at the wake. Since the variation 
of the pressure is given in the first approximation, according to Bernoulli's 
equation, by pUu x = pU d<j>Jdx, it follows that the derivative dtjyjdx must 
also be continuous. The derivative d(j>jdz (the velocity along the wing) is 
in general discontinuous, however. 



§37 ZhukovsMVs theorem 139 

Since the derivative d<f>jdx is continuous, the discontinuity fa-<f>i depends 
only on z, and not on the co-ordinate x along the wake. Thus we have 
the following formula for the lift: 

F y = -puffa-toP*- ( 37 - 2 ) 

The integration over z may be taken over the width of the wake (of course, 
<j>2 — (f)i = outside the wake). 

This formula can be put in a somewhat different form. To do so, we notice 
that, using well-known properties of an integral of the gradient of a scalar, 
we can write the difference fa-fa as a contour integral 

& grade/) -dl = <j> (u y dy+u x dx), 

taken along a contour which starts from the point yi, encircles the body, and 
ends at the point y 2 , thus passing at every point through the region of potential 
flow. Since the wake is thin we can, without changing the integral except by 
quantities of higher order, close this contour by means of the short segment 
from y% to y\. Denoting by T the velocity circulation round the closed 
contour C enclosing the body (Fig. 18), we have 

T = <JJu.dl = ^2-^1, (37.3) 

and for the lift force the formulaf 

F y = -pUJTdz. (37.4) 

The relation between the lift and the circulation given by this formula 
constitutes Zhukovskii's theorem, first derived by N. E. Zhukovskii in 19064 

PROBLEMS 

Problem 1 . Determine the manner of widening of the turbulent wake formed in transverse 
flow past a cylinder of infinite length. 

Solution. The drag/x per unit length of the cylinder is of the order of pUu Y. Combining 
this with the relation (36.1), we find the width Y of the wake to be 

Y = AVixfcJpU*), (1) 

where A is a constant. The mean velocity u in the wake falls off in accordance with 
u ~ Vifxlpx). The Reynolds number R ~ Yufv ~f x }pUv is independent of x, and there 
is therefore no laminar wake. 



t The sign of the velocity circulation is always chosen to be that obtained for a counter-clockwise 
path. The sign in formula (37.3) also depends on the chosen direction of flow. We always suppose 
that the flow is in the positive direction of the #-axis (from left to right). 

% Cf. §46 for the application of this theorem to streamlined wings. 



140 Turbulence §38 

We may mention that, according to experimental results, the constant coefficient in (1) 
is A = -93 ( Y being the half-width of the wake ; if Y is taken as the distance at which the 
velocity u x falls to half its maximum value (at the centre of the wake), then A — 0-41). 

Problem 2. Determine the flow outside the wake formed in transverse flow past a body 
of infinite length. 

Solution. Outside the wake we have potential flow; we shall denote the potential by <b 
to distinguish it from the angle <fi in the system of cylindrical co-ordinates which we take, 
with the z-axis along the length of the body. As in §21, Problem 2, we conclude that we must 
have 

cfu-df = jgjrad<!>'df = f x /pU, 

where now the integration is over the surface of a cylinder of large radius and unit length with 
its axis in the a-direction, and f x is the drag per unit length of the body. The solution of the 
two-dimensional Laplace's equation A * = that satisfies this condition is O = (f x J2npU) log r 
Next, we have for the lift, by formula (37.2), /„ = pU^-O,). The solution of Laplace's 
equation that diminishes least rapidly with distance and has a discontinuity on the plane 
I = is <D = constant X <f>\ since 4>2~<t>i = 2tt, the constant is -fyjlirpU. The flow is given 
by the sum of these two solutions, i.e. 

The cylindrical components of the velocity u are 

u r = d<b\dr = UllirpUr, U<f> = (l/r)a<J>/c# = -/ y /2*y>tfr. (2) 

The velocity u is at a constant angle tan -1 (f y jf x ) to the r-direction. 

Problem 3. Determine the manner of bending of the wake behind a body of infinite length 
when there is a lift force. 

Solution. If there is a lift force, the wake (regarded as a surface of discontinuity) is curved 
in the ary-plane. The function y = y(x) which determines this is given by the equation 
dxJ(ux + U) = djy/tty. Substituting, by (2) of Problem 2, u y « -fyll-npVx and neglecting 
u x in comparison with U, we obtain 

dy/dx = -fyllirpUtx, 
whence 

y = constant -(f y /27rpU 2 ) log x. 

§38. Isotropic turbulence 

We have already mentioned in §33 the particular case of turbulent flow 
that is completely homogeneous and isotropic, the mean velocity being zero 
throughout the fluid. Such a flow may be imagined as that of a fluid which is 
vigorously stirred and then left to itself. The motion decays with time, of 
course. 

The further investigation of isotropic turbulence, and in particular the 
determination of the manner of its decay with time, is based on a conservation 
law first derived by L. G. Loitsyanskii (1939). This law, which holds only 
for isotropic turbulence, is a consequence of the general law of conservation of 
angular momentum, and may be derived as follows. 



§38 Isotropic turbulence 141 

Let us isolate some fairly large volume in an unbounded fluid, and consider 
the total fluid angular momentum M contained in this volume. M has some 
random value, which is not in general zero. On account of the interaction with 
the surrounding regions, M does not remain strictly constant. However, 
since the interaction is a surface effect, it is clear that the time T during which 
M varies appreciably must increase with the dimension L of the volume 
selected. The time T and the dimension L may be arbitrarily large, and in 
this sense the angular momentum M is conserved. 

For convenience in what follows, we suppose that the chosen volume of 
fluid is enclosed in a vessel with fixed solid walls; it is evident that the boun- 
dary conditions at the surface of a very large volume cannot have any effect 
on the volume properties of the flow, in which we are interested. 

According to the general definition, the tensor Mm, which is the total 
angular momentum, is equal to the integral 

p J (xiV k - x k Vi)dV 

taken over the whole volume. We transform this integral as follows : 

f x k VidV = f (xiX k vi)dV- f xtxt—dV- f XiV k dV. 

J J dxi J oxi J 

The first integral on the right-hand side, on being converted into a surface 
integral, is seen to be zero, since the normal velocity component at the walls 
bounding the fluid is zero, so that vjtdfy = v-nd/ = 0. The second integral 
is zero if the fluid is incompressible (div v = 0). Thus 



and we can write 



j x k VidV = — J xivjcdV, 



Mm = 2/> J XiVkdV. 



The sum of the squared components of Ma is equal to twice the squared 
absolute magnitude of the angular momentum vector 

M = p j r xvdF. 
We therefore have 

Af2 = 2/> 2 [ J* XiV k dVf. 
The squared integral can be written as a double integral: 
M* = 2p2 J* J* Xix'iVw'kdVdV. 



142 Turbulence §38 

Finally, we notice that this expression may be rewritten 

m = -P 2 // (xi-x't) 2 Vkv'kdVdV; (38.1) 

the integrals containing the squares xt 2 and x\ 2 vanish, since 

// x'^v k v' k dVAV' = J x'WkdV j v k dV, and jv k dV = 

because the total linear momentum of an incompressible fluid in a fixed 
vessel is zero. 

The factor v k v' k = vv' in the integrand of (38.1 is the scalar product 
of the velocities at two points having co-ordinates ^)and x' k , at a distance 
r = VK**-*'*) 2 ] apart. We average this product over all positions of 
the points x k and x' k (for given r) in the volume concerned: this averaging 
is the same as the one used in §33 in defining the correlation functions. Since 
the flow is isotropic, the quantity v^v' is a function of r only. It falls off 
rapidly with increasing r, since the velocities of the turbulent flow at two 
points a great distance apart may be supposed statistically independent: 
the mean value of their product then reduces to the product of the mean values 
of the individual velocities, which is zero (the mean velocity being everywhere 
zero in the flow under consideration). 

Effecting this averaging under the integral sign in (38.1), we find 

= P 2 jfdV, where / = - JvV r 2 dV. (38.2) 



M 2 



The integrand in / diminishes rapidly with increasing r, so that the integral 
converges; this means that, as the dimension L of the region tends to infinity, 
/ tends to a finite limit. Since the flow is homogeneous,! the quantity / 
is constant everywhere in the fluid, and we can write simply M 2 = p 2 /V. 
We may point out that the angular momentum is thus found to increase as the 
square root of the volume of moving fluid, and not proportionally to the 
volume. This is because the total angular momentum is the sum of a large 
number of statistically independent components (the angular momenta of 
various small portions of the fluid) whose mean values are not zero. 

Thus we conclude that, for isotropic turbulence, the constancy of M implies 
the condition 



f 



v-v'r 2 dV = constant. (38.3) 

This is LoltsyanskiVs law.% 



t Throughout the region, except for a very small part near the surface. 

% Doubts have recently been expressed more than once concerning the applicability of the con- 
servation law (38.3), on account of the behaviour of the velocity correlation at very large distances; 
for example, if this correlation does not decrease sufficiently rapidly, the integral (38.3) may diverge. 
The whole subject seems to be as yet somewhat unclear. 



§38 Isotropic turbulence 143 

The integrand in (38.3) is noticeably different from zero in a region whose 
dimensions are of the order of the scale / of the turbulence (the volume of 
the region ~ Z 3 ), and is there of the order of v 2 l 2 . Hence we have from 
(38.3) 

v 2 l 5 _ constant. (38.4) 

Using this relation, we can determine the manner of the time decay of 
isotropic turbulence. To do so, we estimate the time derivative of the kinetic 
energy of unit volume of the fluid. On the one hand, it may be written as 
being of the order of pv 2 jt. On the other hand, it must equal the energy 
dissipated in unit volume per unit time. According to formula (31.1), 
pe ~ pv*]l (the characteristic velocity here being v). If the two expressions 
are comparable, we find 

/ - vt. (38.5) 

Substituting (38.5) in (38.4), we see that 

v = constant/t 5/7 . (38.6) 

Thus the velocity in isotropic turbulence decays with time inversely as 
* 5/7 . For / we have 

/= constant xt 2 ", (38.7) 

i.e. the external scale of the turbulence increases as t 211 (A. N. Kolmogorov, 
1941). 

According to formulae (38.6) and (38.7), the Reynolds number R ~ vl/v 
decreases as t~ 3n , and after a sufficient time it becomes so small that the 
viscosity begins to be important. The energy dissipation is then determined, 
on the one hand, by the usual formula (16.3), which gives 

/ dvi dvic \ 2 w 2 
* = M T~ + -r- 



\ dxjc dxt 1 I 2 

and, on the other hand, by e ~ v 2 /t. Comparing, we obtain 

/ ~ VH ( 38 - 8 ) 

and then from (38.4) we have 

v = constant/^/4. (38.9) 

These formulae, which are due to M. D. Millionshchikov (1939), give 
the manner of decay of isotropic turbulence in the final period, when the 
effect of the viscosity becomes predominant. 

Isotropic turbulent flow can be brought about by passing a stream through 
a grid having a large number of regularly spaced openings. We denote by U 
the velocity of the original flow, taking the #-axis in the direction of U, and 
the true velocity by U+v, so that v is the velocity of the turbulent flow 
in which we are interested. If we introduce a frame of reference moving 



144 Turbulence §38 

with velocity U, then relative to this frame the fluid executes a turbulent 
flow with velocity v. As we move away from the grid, the averaged turbulent 
flow (with velocity u = v) decays faster than the fluctuating flow. This is 
because the averaged flow has a scale of the order of the dimension a of the 
grid openings, and these, as we shall see, are small in comparison with the 
scale of the fluctuating flow. Consequently, at sufficiently large distances x 
from the grid, the averaged velocity u is almost zero, and the turbulent 
velocity v is just the fluctuating velocity. At such distances the turbulence 
may be regarded as completely isotropic over regions small compared with * 
(though not necessarily small compared with the external scale of the tur- 
bulence). The time decay of the turbulence in the moving frame of reference 
corresponds to a decay with increasing distance from the grid in the original 
stationary frame. The manner of this decay is given by the formulae derived 
above, in which we need only replace / by xjU. Bearing in mind that, at 
distances from the grid of the order of a (the dimension of the openings), 
we must have / ~ a, we can rewrite formula (38.7) as / ~ a{xja) 2n . For the 
velocity we have by (38.5), v ~ lUjx, whence v ~ U{ajx)$ n . 

PROBLEM 

Using equa tion (3 3.17), obtain for isotropic turbulence the quantitative law of decay of 
the quantities v ri Vr2 in the period when viscosity is important (L. G. LoiTSYANSKii, M. D. 

MILLIONSHCHIKOV). 

Solution. In this case we can neglect the B rrr term in (33.17), as being of a higher order 
in the (small) velocity. Introducing the quantity 



2v 8 I 8brr\ 

- — — (r4— : = o. 

r 4 8r \ 8r J 



brr = V r iVr2 = %V 2 -\B r 

(see (33.16)), we obtain for it the equation 

8b rr 2v 8 / _8b r 

~8t 

The solution of this equation that is of interest is 

b rr = constant xe- rV8 "Vi 5/2 ; 

cf. the analogous solution (51.6) of the equation of heat conduction. This gives the asymp- 
totic form of the function b rr for initial conditions such that b rr is any function which de- 
creases sufficiently rapidly with increasing r (just as (51.6) gives the asymptotic law of pro- 
pagation of heat which at the initial instant is concentrated in a small region of space). 



CHAPTER IV 

BOUNDARY LAYERS 

§39. The laminar boundary layer 

We have several times mentioned the fact that very large Reynolds numbers 
are equivalent to very small viscosities, and consequently a fluid may be 
regarded as ideal if R is large. However, this approximation can never be 
used when the flow in question occurs near solid walls. The boundary con- 
ditions for an ideal fluid require only the normal velocity component to vanish; 
the component tangential to the surface in general remains finite. For a 
viscous fluid, however, the velocity at a solid wall must vanish entirely. 

From this we can conclude that, for large Reynolds numbers, the decrease 
of the velocity to zero occurs almost exclusively in a thin layer adjoining the 
wall. This is called the boundary layer, and is thus characterised by the pres- 
ence in it of considerable velocity gradients. The flow in the boundary layer 
may be either laminar or turbulent. In this section we shall consider the 
properties of the laminar boundary layer. The boundary of the layer is not, 
of course, sharp ; the transition from the laminar flow in it to the main stream 
of fluid is continuous. 

The rapid decrease of the velocity in the boundary layer is due ultimately 
to the viscosity, which cannot be neglected even if R is large. Mathemati- 
cally, this appears in the fact that the velocity gradients in the boundary 
layer are large, and therefore the viscosity terms in the equations of motion, 
which contain space derivatives of the velocity, are large even if v is smalL 
The mathematical theory of the boundary layer is due to L. Prandtl. 

Let us derive the equations of motion of the fluid in a laminar boundary 
layer. For simplicity, we consider two-dimensional flow along a plane por- 
tion of the surface. This plane is taken as the xsr-plane, with the #-axis in 
the direction of flow. The velocity distribution is independent of z, and the 
velocity has no ^-component. 

The exact Navier-Stokes equations and the equation of continuity are then 

dv x dv x I dp / d*v x d*v x \ 

V x 1- V v = h V\ 1 — , (39.1) 

x dx Ijy pSx \ 8x* 8y* J K 

8Vy 8Vy \dp (d*Vy 8*Vy\ 

x dx *ly pSy \ 8x* 8y* I V 

8?)r 8v v 

_1 + _Z = o. (39.3) 

dx dy 

145 



146 Boundary Layers §39 

The flow is supposed steady, and the time derivatives are therefore omitted. 

Since the boundary layer is thin, it is clear that the flow in it takes place 
mainly parallel to the surface, i.e. the velocity v y is small compared with v x 
(as is seen immediately from the equation of continuity). 

The velocity varies rapidly along the y-axis, an appreciable change in it 
occurring at distances of the order of the thickness S of the boundary layer. 
Along the #-axis, on the other hand, the velocity varies slowly, an appreciable 
change in it occurring only over distances of the order of a length / charac- 
teristic of the problem (the dimension of the body, say). Hence the y-deriva- 
tives of the velocity are large in comparison with the ^-derivatives. It follows 
that, in equation (39.1), the derivative 8 2 v x /dx 2 may be neglected in compari- 
son with d 2 v x \dy 2 \ comparing (39.1) with (39.2), we see that the derivative 
dp I By is small in comparison with dpjdx (the ratio being of the same order 
as Vyjvx). In the approximation considered we can put simply 

tylty = 0, (39.4) 

i.e. suppose that there is no transverse pressure gradient in the boundary 
layer. In other words the pressure in the boundary layer is equal to the pres- 
sure p(x) in the main stream, and is a given function of x for the purpose of 
solving the boundary-layer problem. In equation (39.1) we can now write, 
instead of dpjdx, the total derivative dp(x)/dx; this derivative can be ex- 
pressed in terms of the velocity U(x) of the main stream. Since we have 
potential flow outside the boundary layer, Bernoulli's equation, p + ^pU 2 
= constant, holds, whence {\jp)dpjdx = - UdJJjdx. 

Thus we obtain the equations of motion in the laminar boundary layer in 
the form 

dv x 8v x d 2 v x 1 dp 

v x H v y v = 

8x dy dy 2 p dx 

dU 



= U- , 

dx' (39.5) 



dv x dvy 
+ — - = 0. 

dx dy 



It can easily be shown that these equations, though derived for flow along a 
plane wall, remain valid in the more general case of any two-dimensional flow 
(transverse flow past a cylinder of infinite length and arbitrary cross-section). 
Here x is the distance measured along the circumference of the cross-section 
from some point on it, andy is the distance from the surface. 

Let Uo be a velocity characteristic of the problem (for example, the 
velocity of the main stream at infinity). Instead of the co-ordinates x, y 
and the velocities v x , v y , we introduce the dimensionless variables x', y', 
v' x , v' y : 

x = lx', y = ///-v/R, v x = Uqv' x , v y = Uov'y/^R (39.6) 



§39 The laminar boundary layer 147 

(and correspondingly U = UoU'), where R = UqIJv. Then the equations 
(39.5) take the form 

, fa'x , fa'y 8V, &U' 

(39.7) 
dv' x fa'y 

+ — - = 0. 



dx' by' 

These equations (and the boundary conditions on them) do not involve the 
viscosity. This means that their solutions are independent of the Reynolds 
number. Thus we reach the important result that, when the Reynolds number 
is changed, the whole flow pattern in the boundary layer simply undergoes 
a similarity transformation, longitudinal distances and velocities remaining 
unchanged, while transverse distances and velocities vary as lfs/K. 

Next, we can say that the dimensionless velocities v' x , v' y obtained by 
solving equations (39.7) must be of the order of unity, since they do not 
depend on R. The same is true of the thickness 8 of the boundary layer in 
terms of the co-ordinates x' y y'. From formulae (39.6) we can therefore 
conclude that 

Vy ~ Ob/VR. ( 39 « 8 ) 

i.e. the ratio of the transverse and longitudinal velocities is inversely propor- 
tional to \/R, and that 

S ~ //VR, (39.9) 

i.e. the thickness of the boundary layer diminishes with increasing Reynolds 
number as l/y^R- 

Let us apply the equations for the boundary layer to the case of plane- 
parallel flow along a flat plate. Let the plane of the plate be the xz half-plane 
with x > (the leading edge of the plate thus being the line * = 0). We 
suppose the plate to extend indefinitely in the positive ^-direction. The 
velocity of the main stream in this case is evidently constant (U = constant). 
The equations (39.5) become 

fax fax &*>x fax , fay n mim 

v x — + v y = v — -, + — = 0. (39.10) 

dx dy By* dx dy 

The boundary conditions at the surface of the plate are that both velocity 
components should vanish: v x = v y = for y = 0, x ^ 0. As we move 
away from the plate, the velocity must approach asymptotically the velocity 
U of the incident flow, i.e. v x = U for y -> ± oo. In the solution of the 
equations for the boundary layer, as we have seen, v x jU and v y <s/{HUv) can 
be functions only of x' = xjl and y' = y\/(Ullv). In the problem under 
consideration, however, the plate is infinite in extent and there are no charac- 
teristic lengths /. Hence v x jU can depend only on a combination of x' and y' 



148 Boundary Layers §39 

which does not involve /, namely y'j\/x' = y-\Z(Ufvx). Similarly, the product 
v'yV x ' must be a function of y /\/x'. Thus we can seek a solution in the form 

** = Uf[yV(Ulvx)l v y = ViUvlxfilyy/iU/vx)], (39.11) 

where/ and /i are some dimensionless functions. Using the second equation 
(39.10), we can express /i in terms of/. The problem thus reduces to the 
determination of a single function / of a single variable | = y\/{UJvx).\ 
In what follows we shall be interested only in the distribution of the 
longitudinal velocity v x (since v y is small). We can draw an important 
conclusion from formula (39.11) without even determining the function/. 
The velocity v x increases from zero at the surface of the plate to a definite 
fraction of Ufor a given value of the argument of/, i.e. for y y/{ U/vx) = any 
given constant. Hence we can conclude that the thickness of the boundary 
layer in flow along a plate is given in order of magnitude by 

8 ~ V(vx/U). (39.12) 

Thus, as we move away from the edge of the plate, 8 increases as the square 
root of the distance from the edge. 

The function / can be determined by numerical integration. A graph of 
this function is shown in Fig. 19. We see that / tends very rapidly to its 
limiting value of unity. J 

The frictional force on unit area of the surface of the plate is 

o xy = 7]{dv x jdy) y= Q. 

A numerical calculation gives 

o xy = 0-332V(^C/ 3 /^). (39.13) 

If the plate is of length / (in the x-direction), then the total frictional force 
on it per unit length in the ^-direction is 

i 
F = 2 a X ydx. 
o 

The factor 2 is due to the fact that the plate has two sides exposed to the 



t It is easily shown that, if the function <£(£) is such that/(£) = <f>'(£), then/ 1 (f) = 4(£<£'— <f>), 
while (f> satisfies the equation <fxf>"+2<l>'" = 0, with the boundary conditions <j> = </>' = for £ = 0, 
</>' = 1 for £ = oo. 

X The "displacement thickness" 8*, sometimes used to characterise the thickness of the boundary 
layer, is defined by 

00 

j (U-v x )dy = US*. 
o 

It is equal to 1*72 \/(vx/U). 



§39 



The laminar boundary layer 



149 



fluid. Substituting (39.13), we have 

F = 1-328 VMC/ 8 ) ( 39 - 14 ) 

(H. Blasius, 1908). We may point out that the frictional force is proportional 
to the f power of the velocity of the main stream. Formula (39.14) can be 
applied only to fairly long plates, for which the Reynolds number Uljv is 
fairly large. The force is customarily expressed in terms of the drag coefficient, 
defined as the dimensionless ratio 

C = FfoW.Zl. (39.15) 

By (39.14), this quantity, for laminar flow along a plate, is inversely pro- 
portional to the square root of the Reynolds number: 

C = 1-328/VR. C 39 - 1 ^) 

The quantitative formulae obtained above relate, of course, only to flow 

along a flat plate. The qualitative results, however, such as (39.8) and (39.9), 

hold for flow past bodies of any shape; in such cases / is the dimension of 

the body in the direction of flow. 



1-0 
0-8 
0-6 
0-4 
0-2 



3 4 5 



Fig. 19 



We may make special mention of two cases of the boundary layer. If we 
have a plane disk, of large radius, rotating in the fluid about an axis perpen- 
dicular to its plane, then to estimate the thickness of the boundary layer we 
must replace U in (39.12) by Q.x, where Q is the angular velocity of rotation. 
We then find 

S ~ -\/(v/Q). (39.17) 

We see that the thickness of the boundary layer may be regarded as a constant 
over the surface of the disk, in accordance with the exact solution of this 
problem obtained in §23. The magnitude of the frictional forces on the disk, 
as obtained from the equations for the boundary layer, is of course (23.4), 
since this formula is exact and therefore holds for laminar flow with any 
value of R. 



150 Boundary Layers §39 

Finally, let us consider the laminar boundary layer formed at the walls of 
a pipe near the point of entry of fluid. The fluid usually enters the pipe with 
a velocity distribution which is almost constant over the cross-section, and 
the velocity falls to zero entirely within the boundary layer. As we move 
away from the entrance to the pipe, the fluid layers nearer the axis are re- 
tarded. Since the mass of fluid that passes each cross-section is the same, the 
inner part of the stream, where the velocity is still uniform, must be 
accelerated as its diameter is reduced. This continues until a Poiseuille 
velocity distribution is asymptotically reached; this distribution is thus found 
only at some distance from the entrance to the pipe. It is easy to determine 
the order of magnitude of the length / of the "inlet section". It is given by the 
fact that, at a distance / from the entrance, the thickness of the boundary 
layer is of the same order of magnitude as the radius a of the pipe, so that the 
boundary layer fills almost the whole cross-section. Putting in (39.12) 
x ~ / and S ~ a, we obtain 

/ ~ a 2 Ujv ~ aR. (39.18) 

Thus the length of the inlet section is proportional to the Reynolds number, f 

PROBLEMS 

Problem 1 . Determine the thickness of the boundary layer near a stagnation point (see §10). 

Solution. Near the stagnation point the fluid velocity (outside the boundary layer) is 
proportional to the distance x from that point, so that we can put U — ex. By estimating the 
magnitudes of the terms in the equations (39.5) we find S ~ \/(v/c). Thus the thickness of 
the boundary layer near the stagnation point is finite (and, in particular, does not vanish at 
the stagnation point itself). 

Problem 2. Determine the flow in the boundary layer in a converging channel between 
two non-parallel planes (K. Pohlhausen, 1921). 

Solution. Considering the boundary layer along one of the planes, we measure the co- 
ordinate x along that plane from the point O (Fig. 8, §23). For an ideal fluid we should have 
the velocity U = Qfoucp ; the corresponding pressure gradient is, by Bernoulli's equation, given 
by 

Id* d O 2 

-T"= --(|C/2)= * 



p dx dx a 2 # 3 /o 2 

It is easy to see that v x and v y must be sought in the form 

v x = (Qlpooc)f(y/x), v y = {Qlp«x)fi(ylx). 

From the equation of continuity we obtain f t = (y/x)f, and the first equation (39.5) then gives 
for the function / 

(W0/"=l-/2, 
where the prime denotes the differentiation of/ with respect to its argument £ = y/x. The 



f We shall not discuss the theory of the boundary layer for a compressible fluid, which is, of course, 
considerably more complicated than that for an incompressible fluid. An account of this theory may 
be found in: N. E. Kochin, I. A. Kibel' and N. V. Roze, Theoretical Hydromechanics (Teoreticheskaya 
gidromekhanika), Part 2, 3rd ed., Chapter II, §§35, 36, Moscow 1948; H. Schlichting, Boundary 
Layer Theory, Pergamon Press, London 1955; L. Howarth ed., Modern Developments in Fluid 
Dynamics: High Speed Flow, vol. 1, Oxford 1953. 



§40 Flow near the line of separation 151 

boundary conditions are /(0) = 0, /(co) = 1 (since we must have (v x )y=f> = 0, (v x )y=(o 
= Qlpcuc). A first integral of the equation is 

(vap/20)/' 2 =/-£/3+ constant. 

Since /tends to unity as y -*■ oo, we see that/' tends to a definite limit, which can only be 
zero. The constant being thereby determined, we find 

(mp/2£)/'2 = _ K /-l)2( /+2 ). 

Since the right-hand side is always negative for ^ / < 1, we must have Q < 0. That is, 
a boundary layer of the type in question is formed only by flow in a converging channel 
(and only at large Reynolds numbers R = \Q\/vp), and not by flow in a diverging channel, in 
accordance with the results of §23. Integrating again, we have finally 

/ = 3 tanh2[log( V2 + VV + fl/W 2 ")] - 2- 

§40. Flow near the line of separation 

In describing the line of separation (§34) we have already mentioned that 
the actual position of this line on the surface of the body is determined by the 
properties of the flow in the boundary layer. We shall see below that, from a 
mathematical point of view, the line of separation is a line whose points 
are singular points of the solutions of (PrandtPs) equations of motion in the 
boundary layers. The problem is to determine the properties of these solu- 
tions near such a line of singularities.! 

We know already that, from the line of separation, there begins a surface 
which extends into the fluid and marks off the region of turbulent flow. The 
flow is rotational throughout the turbulent region, whereas in the absence of 
separation it would be rotational only in the boundary layer, where the vis- 
cosity is important; the vorticity would be zero in the main stream. Hence 
we can say that separation causes the vorticity to "penetrate" from the boun- 
dary layer into the fluid. By the conservation of circulation, however, 
this "penetration" can occur only by the direct mixing of fluid moving near 
the surface (in the boundary layer) with the main stream. In other words, the 
flow in the boundary layer must be separated from the surface of the body, the 
streamlines consequently leaving the surface layer and entering the interior 
of the fluid. This phenomenon is therefore called separation or separation of 
the boundary layer. 

The equations of motion in the boundary layer lead, as we have seen, to the 
result that the tangential velocity component (v x ) in the boundary layer is 
large compared with the component (%) normal to the surface of the body. 
This relation between v x and v y derives from our basic assumptions regarding 
the nature of the flow in the boundary layer, and must necessarily be found 
wherever Prandtl's equations have physically meaningful solutions. Mathe- 
matically, it is found at all points not lying in the immediate neighbourhood 
of singular points. But if v y <^ v x it follows that the fluid moves along the 



f The treatment of the problem given here, due to L. D. Landau, is somewhat different from 
that usually given. 



152 Boundary Layers §40 

surface of the body, and moves away from the surface only very slightly, 
so that there can be no separation. We therefore reach the conclusion that 
separation can occur only on a line whose points are singularities of the 
solution of PrandtPs equations. 

The nature of these singularities also follows immediately. For, as we 
approach the line of separation, the flow deviates from the boundary layer 
towards the interior of the fluid. In other words, the normal velocity com- 
ponent ceases to be small compared with the tangential component, and is 
now of at least the same order of magnitude. We have seen (cf. (39.8)) 
that the ratio v y jv x is of the order of Ij-y/R, so that an increase of v y to the 
point where v y ~ v x means an increase by a factor of y'R. Hence, for suffi- 
ciently large Reynolds numbers (which, of course, we are considering) 
we may suppose that v y increases by an infinite factor. If we use Prandtl's 
equations in dimensionless form (see (39.7)), the situation just described is 
formally equivalent to an infinite value of the dimensionless velocity v' y 
on the line of separation. 

In order to simplify the subsequent discussion a little, we shall consider 
the two-dimensional problem of transverse flow past a body of infinite length. 
As usual, x is the co-ordinate along the surface in the direction of flow, while 
y is the distance from the surface of the body. Instead of a line of separation, 
we now have a point of separation, namely the intersection of the line of 
separation with the xy-plane; in the co-ordinates used, this is the point 
x = constant s xo, y = 0. Let x < xo be the region in front of the point 
of separation. 

According to the above results, we have for allf y 

v y (xo,y) = oo. (40.1) 

In Prandtl's equations, however, v y is a kind of parameter, which is usually 
of no interest (on account of its smallness) in investigating the flow in the 
boundary layer. Hence it is necessary to ascertain the properties of the func- 
tion v y near the line of separation. 

It is clear from (40.1) that, for x = xo, the derivative dv y J8y also becomes 
infinite. From the equation of continuity, dv x \dx-\-dv y \dy = 0, it then fol- 
lows that (dv x /8x) x==Xo is infinite, or dxjdv x = 0, where x is regarded as a 
function of v x and y. We denote by ©o(y) the value of the function v x (x, y) 
for x = xq: vo(y) = v x (xq, y). Near the point of separation, the differences 
v x — vo and xq — x are small, and we can expand xq — x in powers of v x — vo 
(for a given y). Since (dx/dv x ) v=v<) = 0, the first-order term in this expansion 
must vanish identically, and we have as far as terms of the second order 
xo-x =f(y)(v x -v ) 2 , or 

v% = My) + *{y)V( x o - x )> (40.2) 



f Except y = 0, where we must always have v y — in accordance with the boundary conditions 
at the surface of the body. 



§40 Flow near the line of separation 153 

where a = Ijy/f is some function of y alone. Putting now 

foy _ dvx _ afcy) 

dy &v 2\/(#o — #) 

and integrating, we have for v y 

vy = Ky)!V(xo-x)> (40-3) 

where 

is another function of _y. 

Next, we use the first equation (39.5): 

dv x &v x d*v x 1 dp 

a^ + v y = v — — . (40.4) 

dx dy dy 2 p dx 

The derivative d z v x jdy 2 does not become infinite for x = xo, as we see 
from (40.2). The same is true of dp/cbc, which is determined by the flow 
outside the boundary layer. Both terms on the left-hand side of equation 
(40.4) become infinite, however. In the first approximation we can therefore 
write for the region near the point of separation v x dv x jdx-hv y dv x jdy = 0. 
Substituting dv x jdx = — dvy/dy, we can rewrite this as 



8v y dv x d 

Vy-— = V x 2 



- (-)-o- 

By dy 8y \ v x I 

Since the velocity v x does not in general vanish for x = xo, it follows that 
8(vylv x )Jdy = 0, i.e. the ratio v y jv x is independent of y. From (40.2) and 
(40.3), we have to within terms of higher order 

v% woOOVfao-*) 

If this is a function of x alone, we must have fi(y) = %Avo(y), where A is a 
numerical constant. Thus 

Vy . **> , (40.5) 

2-y/(x — x) 

Finally, noticing that a and /S in (40.2) and (40.3) obey the relation a = 2j8', 
we obtain en — A dvojdy, so that 

v x = v {y) + A(dv ldy) \Z(x - x). (40.6) 

Formulae (40.5) and (40.6) determine v x and v y as functions of x near 
the point of separation. We see that each can be expanded in this region in 
powers of \^{xq — x), the expansion of v y beginning with the —1 power, so 



154 Boundary Layers §40 

that v y becomes infinite as (#0 — x)~l for x -> xq. For x > xq, i.e. beyond 
the point of separation, the expansions (40.5) and (40.6) are physically 
meaningless, since the square roots become imaginary; this means that the 
solutions of Prandtl's equations which give the flow up to the point of 
separation cannot meaningfully be continued beyond that point. 

From the boundary conditions at the surface of the body, we must always 
have v x = v y = for y = 0. We therefore conclude from (40.5) and (40.6) 
that 

v (0) = 0, (dvoldy)y =0 = 0. (40.7) 

Thus we have the important result (due to Prandtl) that, at the point of 
separation itself (x = xo, y = 0), not only the velocity v x but also its first 
derivative with respect to y is zero. 

It must be emphasised that the equation dv x /dy = on the line of separa- 
tion holds only when v y becomes infinite for that value of x. If the constant 
A in (40.5) happens to be zero, so that v v {xq, y) # 00, then the point x = xq, 
y = at which the derivative 8v x /8y vanishes would have no other particular 
properties, and would not be a point of separation. A can vanish, however, 
only by chance, and such an event is therefore unlikely. In practice a point 
on the surface of the body at which dv x jdy = is always a point of separation. 

If there is no separation at the point x = xo (i.e. if A = 0), then for 
x > xq we have (8v x /dy) y =o < 0, i.e. v x becomes negative (of increasing 
absolute magnitude) as we move away from the surface, y being still small. 
That is, the fluid beyond the point x = xq moves, in the lower parts of the 
boundary layer, in the direction opposite to that of the main stream ; there is a 
"back-flow" of fluid at this point. It must be emphasised that from such 
arguments we cannot conclude that there is necessarily a point of separation 
where dv x jdy = 0; the whole flow pattern with the "back-flow" might lie 
(as it does for A = 0) entirely within the boundary layer and not enter the 
main stream, whereas it is characteristic of separation that the flow enters 
the main body of the fluid. 

It has been shown in the previous section that the flow pattern in the boun- 
dary layer is similar for different Reynolds numbers, and, in particular the 
scale in the ^-direction remains unchanged. It follows from this that the 
value #0 of the co-ordinate x for which the derivative {dv x jdy) y =o is zero is 
the same for all R. Thus we have the important result that the position of 
the point of separation on the surface of the body is independent of the 
Reynolds number (so long as the boundary layer remains laminar, of course ; 
see §45). 

Let us also ascertain the properties of the pressure distribution p(x) 
near the point of separation. For y = the left-hand side of equation (40.4) 
is zero together with v x and v y , and there remains 

v(^/^ 2 )h = (Vp)dpldx. (40.8) 

It is clear from this that the sign of dpfdx is the same as that of (d 2 v x ]dy z ) y =o. 



§40 Flow near the line of separation 155 

When (dvxldy)y=o > we can say nothing regarding the sign of the second 
derivative. However, since v x is positive and increases away from the wall 
(in front of the point of separation), we must always have (d 2 %/#y 2 )2/=o > 
at x = xo itself, where dv x /dy = 0. Hence we conclude that 

(dp/dx) x = Xa > 0, (40.9) 

i.e. the fluid near the point of separation moves from the lower pressure to the 
higher pressure. The pressure gradient is related to the gradient of the 
velocity U(x) outside the boundary layer by (l/p)dp/d# = — U dU/dx. 
Since the positive direction of the axis is the same as the direction of the 
main stream, U > 0, and therefore 

(dU/dx) x = Xa < 0, (40.10) 

i.e. the velocity U decreases in the direction of flow near the point of separa- 
tion. 

From the results obtained above we can deduce that there must be separa- 
tion somewhere on the surface of the body. For there is on both the front and 
the back of the body a point (the stagnation point) at which the fluid velocity 
is zero for potential flow of an ideal fluid. Consequently, for some value of x, 
the velocity U(x) must begin to decrease, and finally it becomes zero. It is 
clear, however, that the fluid moving over the surface of the body is retarded 
the more strongly, the closer it is to the surface (i.e. the smaller is y). Hence, 
before the velocity U(x) is zero at the outer limit of the boundary layer, the 
velocity in the immediate neighbourhood of the surface must be zero. Mathe- 
matically, this evidently means that the derivative dv x /dy must always vanish 
(and therefore there must be separation) for some x less than the value for 
which U(x) = 0. 

In flow past bodies of any form the calculations can be carried out in an 
entirely similar manner, and they lead to the result that the derivatives 
dv x [dy, dvz/dy of the two velocity components v x and v z tangential to the 
surface of the body vanish on the line of separation (the jy-axis, as before, 
is along the normal to the portion of the surface considered). 

We may give a simple argument which demonstrates the necessity of separa- 
tion in cases where the fluid would otherwise have a rapid increase of pressure 
(and therefore a rapid decrease in the velocity U) in the direction of its flow 
past the body. Over a small distance A.x = x% — xi, let the pressure p 
increase rapidly from/>i top2 (j>2 > Pi)- Over the same distance A#, the fluid 
velocity U outside the boundary layer falls from its initial value U\ to a 
considerably smaller value TJi determined by Bernoulli's equation: 

i(Ul 2 -U2 2 )=(p2-pl){p. 

Since p is independent of y, the pressure increase p<i —p± is the same at all 
distances from the surface. If the pressure gradient dpfdx ~ (j>2—pi)l^x 
is sufficiently high, the termvd^^dy 2 involving the viscosity may be omitted 
from the equation of motion (40.4) (if, of course, y is not small). Then, to 



156 Boundary Layers §41 

estimate the change in the velocity v in the boundary layer, we can use 
Bernoulli's equation, putting %{v2 2 — vi 2 ) = — (P2—pi)lp, or, from the 
equation previously obtained, V2 2 = v± 2 — (U1 2 — U2 2 ). The velocity v\ in 
the boundary layer is less than that of the main stream, and we can select 
a value of y for which vi 2 < Ui 2 — U2 2 . The velocity V2 is then imaginary, 
showing that Prandtl's equations have no physically significant solutions. 
In fact, there must be separation in the distance Ax, as a result of which the 
pressure gradient is reduced. 

An interesting case of the appearance of separation is given by flow at an 
angle formed by two intersecting solid surfaces. For laminar potential flow 
outside an angle (Fig. 3), the fluid velocity at the vertex of the angle would 
become infinite (see §10, Problem 6), increasing in the stream approaching the 
vertex and diminishing in the stream leaving the vertex. In reality, the rapid 
decrease in velocity (and corresponding increase in pressure) beyond the 
vertex would lead to separation, the line of separation being the line of 
intersection of the surfaces. The resulting flow pattern is that discussed in 
§35. 

In laminar flow inside an angle (Fig. 4), the fluid velocity is zero at the 
vertex. In this case the velocity diminishes (and the pressure increases) in 
the flow approaching the vertex. The result is in general the appearance of 
separation, the line of separation being upstream from the vertex of the angle. 

PROBLEM 

Determine the least possible increase Ap in the pressure which can occur (in the main 
stream) over a distance Ax and cause separation. 

Solution. Let y be a distance from the surface of the body at which, firstly, Bernoulli's 
equation can be applied and, secondly, the squared velocity v 2 (y) in the boundary layer is 
less than the change | A[/ 2 | in the squared velocity outside that layer. For v(y) we can write, 
in order of magnitude, v(y) K y dv/dy ~ Uy/S, where S ^ \/(vlJU) is the width of the 
boundary layer and I the dimension of the body. Equating, in order of magnitude, the two 
terms on the right-hand side of equation (40.4), we find 

(l/p)A^/Ax ~ w(y)ly 2 ~ vUjBy. 

From the condition 

v 2 = \AU 2 \ = {2jp)Ap we have U 2 y 2 l8 2 ~ Apjp. 

Eliminating y, we finally obtain 

Ap ~ P U 2 (Axll)K 



§41. Stability of flow in the laminar boundary layer 

Laminar flow in the boundary layer, like any other laminar flow, becomes to 
some extent unstable at sufficiently large Reynolds numbers. The manner 
of the loss of stability in the boundary layer is similar to that which occurs for 
flow in a pipe (§29). 

The Reynolds number for flow in the boundary layer varies over the surface 



§41 Stability of flow in the laminar boundary layer 157 

of the body. For example, in flow along a plate we could define the Reynolds 
number as Ra; = Uxjv, where x is the distance from the leading edge of the 
plate, and U the fluid velocity outside the boundary layer. A more suitable 
definition for the boundary layer, however, is one in which the length para- 
meter directly characterises the thickness of the layer; such, for instance, is the 
"displacement thickness" S* (see the second footnote to §39). We then 
have R§* = U8*jv. Since the dependence of the boundary-layer thickness 
on the distance x is given by formula (39.12), it is clear that R§* ~ V^-t 

Because the change of the layer thickness with distance is comparatively 
slow, it may be neglected in investigating the stability of flow in a small 
portion of the layer, and we may consider a rectilinear two-dimensional flow, 
with a velocity profile which does not vary along the ^-axis.J Then, from a 
mathematical point of view, the problem is entirely analogous to that of 
the stability of flow between two parallel planes, discussed in §29. The only 
difference is in the form of the velocity profile; instead of a symmetrical 
profile with v = on both sides, we now have an unsymmetrical profile in 
which the velocity changes from zero at the surface of the body to some 
given value U, the velocity of the flow outside the boundary layer. The 
investigation leads to the following results (Lin, 1945 ; see C. C. Lin, The 
Theory of Hydrodynamic Stability, Cambridge 1955). 

The form of the limiting curve of stability in the coR-plane (see §29) 
depends on the form of the velocity profile in the boundary layer. If the 
velocity profile has no point of inflexion, and the velocity v x increases 
monotonically with the curve v x = v x {y) everywhere convex upwards (Fig. 
20a), then the boundary of the stable region is completely similar in form to 
that which is obtained for flow in a pipe: there is a minimum value R = R cr 
at which amplified perturbations first appear, and for R -> oo both branches 
of the curve are asymptotic to the axis of abscissae (Fig. 21a). For the velocity 
profile which occurs in the boundary layer on a flat plate, the critical Reynolds 
number is found by calculation to be R§*,cr ~ 420.ft 

A velocity profile of the kind shown in Fig. 20a cannot occur if the fluid 
velocity outside the boundary layer decreases downstream. In this case the 
velocity profile must have a point of inflexion. For, let us consider a small 
portion of the surface, which we may regard as plane, and let x be again the 
co-ordinate in the direction of flow, and y the distance from the wall. From 
(40.8) we have 

v(d 2 v x lfy 2 )y=o = (llp)dpldx = -UdU/dx, 
whence we see that, if U decreases downstream (dUjdx < 0), we must have 
d^x/dy 2 > near the surface, i.e. the curve v x = Vx(y) is concave upwards. 
As y increases, the velocity v x must tend asymptotically to the finite limit U. 



t For example, in a laminar boundary layer on a flat plate R^» = 1 - 72\/R:r. 

j In doing so, of course, we pass over the question of the effect which the curvature of the surface 
may have on the stability of the boundary layer. 

ft For R»* -> oo, co tends to zero, on the two branches I and II of the limiting curve, as R^* - * and 
R^* -1 / 5 respectively. 



158 



Boundary Layers 



§41 



It is then clear from geometrical considerations that the curve must become 
convex upwards, and therefore must have a point of inflexion (Fig. 20b). 
In this case the form of the curve defining the stable region is slightly changed : 
the two branches have different asymptotes for R -► oo, one tending to the 
axis of abscissae and the other to a finite non-zero value of o> (Fig. 21b). 
The presence of a point of inflexion also reduces considerably the value of R cr . 




The fact that the Reynolds number increases along the boundary layer 
makes the behaviour of the perturbations as they are carried downstream 
somewhat unusual. Let us consider flow along a flat plate, and suppose that a 
perturbation of given frequency co occurs at some point in the boundary 
layer. Its propagation downstream corresponds to a movement in Fig. 21a 
to the right along a horizontal line a> = constant. The perturbation is at 
first damped : then, on reaching branch I of the stability curve, it begins to 
be amplified. This continues until branch II is reached, whereupon the 
perturbation is again damped. The total "amplification coefficient" for the 
perturbation during its passage through the region of instability increases 
very rapidly as this region moves towards large R (i.e. as the corresponding 
horizontal segment between branches I and II moves downwards). 

These results, however, do not answer the question whether true absolute 
instability occurs in the laminar boundary layer for sufficiently large R — 
that is, instability due to the amplification in time of perturbations at a given 
point (see §29). As with flow in a pipe, no such investigation has yet been 
made. 



§42 



The logarithmic velocity profile 



159 



The experimental results for flow along a flat plate show that the point 
where turbulence appears in the boundary layerf depends to a considerable 
extent on the intensity of the perturbations in the main stream. For marked 
perturbations, the boundary layer was observed to become turbulent for 
Rs* » 560. As the intensity of the perturbations diminishes, the onset of 
turbulence is postponed to higher values of R s «, which seem to tend to a 
finite limit of about 3000. 




It is possible that the existence of the limit indicates the presence of true 
absolute instability for sufficiently high values of R. On the other hand, 
it may be that, because of the extremely rapid increase of the "amplification 
coefficient" with R, the "displacement" instability of the kind described above 
may give the appearance of true instability. 

§42. The logarithmic velocity profile 

Let us consider plane-parallel turbulent flow along an unbounded plane 

t Because the Reynolds number varies along the plate, the whole boundary layer does not become 
turbulent immediately, but only the part where R^» exceeds a certain value. For a given incident 
velocity, this means that turbulence begins at a definite distance from the leading edge; as the velocity 
increases, this distance approaches zero. 



160 Boundary Layers §42 

surface; the term "plane-parallel" applies, of course, to the time average of the 
flow.f We take the direction of the flow as the ar-axis, and the plane of the 
surface as the xsr-plane, so that y is the distance from the surface. The y and 
z components of the mean velocity are zero: u x = u, u y = u z = 0. There 
is no pressure gradient, and all quantities depend on y only. 

We denote by a the frictional force on unit area of the surface ; this force is 
clearly in the ^-direction. The quantity o- is just the momentum trans- 
mitted by the fluid to the surface per unit time; it is the constant flux of 
the ^-component of momentum, which is in the negative jy-direction, and 
gives the amount of momentum transmitted from the layers of fluid remote 
from the surface to those nearer it. 

The existence of this momentum flux is due, of course, to the presence of a 
gradient, in the jy-direction, of the mean velocity u. If the fluid moved 
with the same velocity at every point, there would be no momentum flux. 
The converse problem can also be stated: given some definite value of a, 
what must be the motion of a fluid of given density p to give rise to a momen- 
tum flux a? For large Reynolds numbers, the viscosity v is, as usual, unim- 
portant; it becomes important only for small distances y (see below). Thus 
the value of the velocity gradient dujdy at each point must be determined by 
the constant parameters p, a and, of course, the distance y itself. The 
dimensions of these quantities are respectively g/cm 3 , g/cm sec 2 and cm. 
The dimensions of the derivative dujdy are 1/sec. The only combination 
of p, a and y that has the right dimensions is Vi^lpy 2 )- Hence we must have 

dujdy = vWrifty. (42.1) 

where b is a numerical constant ; b cannot be calculated theoretically, and 
must be determined experimentally. It is found to bej 

b = 0417. (42.2) 

We introduce the more convenient notation v% = ^{crjp), so that 

a = /w # 2 . (42.3) 

The quantity v% has the dimensions cm/sec and acts as a characteristic velocity 
for the turbulent flow considered; then (42.1) becomes dujdy = v%jby, 
whence 

« = (»#/*)(log^ + c). (42.4) 

where c is a constant of integration. To determine this constant we cannot 
use the ordinary boundary conditions at the surface, since for y = the first 
term in (42.4) becomes infinite. The reason for this is that the above expres- 
sion is really inapplicable at very small distances from the surface, since the 
effect of the viscosity then becomes important, and cannot be neglected. 



f The results given in §§42-44 are due to T. von Karman and L. Prandtl. 

j The value of this constant, and of one in formula (42.8) below, are obtained from measurements 
of the velocity distribution near the walls of a pipe in which there is turbulent flow. 



§42 The logarithmic velocity profile 161 

There are also no conditions at infinity, since for y = oo the expression 
(42.4) again becomes infinite. This is because, in the idealised conditions 
which we have imposed, the surface is unbounded, and its influence therefore 
extends to infinitely great distances. 

Before determining the constant c, we may first point out the following 
important property of the flow considered: contrary to what usually happens, 
it has no characteristic constant parameters of length which might give the 
external scale of the turbulence. This scale is therefore determined by the 
distance y itself: the scale of turbulent flow at a distance j from the surface is 
of the order of y. The fluctuating velocity of the turbulence is of the order of v m . 
This also follows at once from dimensional arguments, since v% is the only 
quantity having the dimensions of velocity which can be formed from the 
quantities or, p, y at our disposal. It should be emphasised that, whereas the 
mean velocity decreases with y, the fluctuating velocity remains of the same 
order of magnitude at all distances from the surface. This result is in accor- 
dance with the general rule that the order of magnitude of the fluctuating 
velocity is determined by the variation Am of the mean velocity (§31). In 
the present case, there is no characteristic length / over which the variation 
of the mean velocity could be taken; Am must now be defined, reasonably, 
as the change in u when the distance y changes appreciably. According to 
(42.4), such a change in y causes a change in the velocity u that is just of the 
order of v m . 

At sufficiently small distances from the surface, the viscosity of the fluid 
begins to be important; we denote the order of magnitude of these distances 
by yo, which can be determined as follows. The scale of the turbulence at 
these distances is of the order of yo, and the velocity is of the order of v m . 
Hence the Reynolds number which characterises the flow at distances of the 
order of jo is R ~ v^yolv. The viscosity begins to be important when R 
becomes of the order of unity. Hence we find that 

yo ~ vjv mt (42.5) 

and this determines yo. 

At distances from the surface small compared with yo, the flow is deter- 
mined by ordinary viscous friction. The velocity distribution here can be 
obtained directly from the usual formula for viscous friction : a = pv dujdy, 
whence 

u = ayjpv = v m 2 y/v. (42.6) 

Thus, immediately adjoining the wall, there is a thin layer of fluid in which 
the mean velocity varies linearly with y; the velocity is small throughout 
this layer, varying from zero at the surface itself to values of the order of 
v m for y ~ yo. We shall call this layer the viscous sublayer. 

It must be emphasised that the flow here is turbulent, and in this respect 
the customary name "laminar sublayer" is unsuitable. The resemblance to 
laminar flow lies only in the fact that the mean velocity is distributed accord- 
ing to the same law as the true velocity would be for laminar flow under the 



162 Boundary Layers §42 

same conditions. There is, of course, no sharp boundary between the viscous 
sublayer and the remainder of the flow, and the concept of the viscous sub- 
layer is therefore to some extent qualitative. 

The longitudinal component v' x of the fluctuating velocity in the viscous 
sublayer is of the same order of magnitude as the mean velocity, and in 
particular is proportional to y (~ v^yjy ). It therefore follows from the 
equation of continuity that the derivative dv'yjdy = - dv' x Jdx is proportional 
to y, and so the transverse component v' y of the fluctuating velocity varies as 
y 2 ( ~ v *y 2 ly<?)' Next, it follows from the linearity of the equations of motion 
in the viscous sublayer (the non-linear terms being there small compared with 
the viscosity terms) that the periods of the turbulent eddies are the same 
throughout the thickness of the sublayer. Multiplying these periods by the 
fluctuating velocity, we find that the longitudinal distances traversed by the 
fluid particles in their fluctuating motion are proportional to y, in order of 
magnitude, and the transverse distances are proportional to y*(~ y 2 /yo)- 

We shall not be further interested in the flow in the viscous sublayer. 
Its presence has to be taken into account only in making the appropriate 
choice of the constant of integration in (42.4). This constant must be chosen 
so that the velocity becomes of the order of v m at distances of the order of yo. 
For this to be so, we must take c = -log^o, so that u = (vjb) log(y[y ), or 

u = (v m fb) log(yv#/ v ). (42.7) 

This formula determines (for a certain range of y) the velocity distribution in 
the turbulent stream which flows along the surface. This distribution is 
called the logarithmic velocity profile. 

The argument of the logarithm in formula (42.7) should include a numeri- 
cal coefficient. However, in the formulae which we shall derive we shall 
require only "logarithmic" accuracy. This means that the argument of the 
logarithm is supposed large, and we neglect not only terms proportional to 
lower powers of the argument but also those involving the logarithm to lower 
powers than in the principal term. The introduction of a small numerical 
coefficient in the argument of the logarithm in (42.7) is equivalent to adding 
a term of the form constant xv m , where the constant is of the order of unity; 
this term does not contain the logarithm, and therefore we neglect it. How- 
ever, it must be borne in mind that the argument of the logarithm in the 
formulae derived here is not so large that its logarithm is also very large, and 
so the accuracy of the formulae is not very high. 

These formulae can be made more exact by introducing a numerical 
coefficient in the argument of the logarithm, or, what is the same thing, adding 
a constant to the logarithm. These constants, however, cannot be calculated 
theoretically, and have to be determined from experimental results. For 
example, a more exact formula for the velocity distribution can be written 
in the form 

u = z; # [240 log0w # /v) + 5-84]. (42.8) 



§43 Turbulent flow in pipes 163 

It is not difficult to determine the energy dissipation e per unit mass of 
fluid, a is the mean value of the component Ilgy of the momentum flux 
density tensor II tt = pvtv k - r)(dv t ldx k + dv k \dxi). Outside the viscous sub- 
layer, the viscosity term may be omitted, so that a = pv x v y . Introducing 
the fluctuating velocity v', we can write v x = u + v' x ; the velocity v y is 
itself the fluctuating velocity v' y , since its mean value is zero. The result is 



O = pV X Vy = pV' X v'y + pUV'y = pv' X V'y. 

Next, the energy flux density in the y-direction is (p + $pv 2 )vy, the viscosity 
term being again omitted. Putting in the second term 

v* = (u+v'xf+v'tpWf 

and averaging, we obtain 



pv'y + lp(v' X 2 v'y + V'y* + v' gV y) + p UV'Wy. 

Here only the last term need be retained. The reason is that the fluctuating 
velocity is of the order of © # , and hence, to logarithmic accuracy, it is small 
compared with «. The turbulent fluctuations of the pressure p are of the 
order of pv m 2 (cf. (31.4)), and so we can, to the same accuracy, neglect the 
corresponding term in the energy flux. Thus we have for the mean energy 
flux density puv' x v' y = ua. As we approach the surface, this flux decreases, 
because the energy is dissipated. The decrease in the energy flux density 
on approaching the surface by a distance d> is <r(dw/dy)d>. This is the amount 
of energy converted into heat in a fluid layer of thickness dy and of unit area. 
Hence we conclude that the energy dissipation per unit mass is (ojp)Auj6y, 
or 

e = vflby = {ajpflby. (42.9) 

§43. Turbulent flow in pipes 

Let us now apply the above results to turbulent flow in a pipe. Near the 
walls of the pipe (at distances small compared with its radius a), the surface 
may be approximately regarded as plane, and the velocity distribution must 
be given by formula (42.7) or (42.8). Since the function logjy varies only 
slowly, we can use formula (42.7) to logarithmic accuracy to give the mean 
velocity U of the flow in the pipe if we replace y in that formula by a : 

U = (v*/b)\og(avM. (43.1) 

By U we mean the volume of fluid that passes through a cross-section of the 
pipe per unit time, divided by the cross-sectional area: U = Qlpna 2 . 

In order to relate the velocity U to the pressure gradient Apjl which 
maintains the flow (A/> being the pressure difference between the ends of 
the pipe, and / its length), we notice that the force on a cross-section of the 



164 Boundary Layers §43 

flow is 7ra 2 Ap. This force overcomes the friction at the walls. Since the 
frictional force per unit area of the wall is a = pv m 2 , the total frictional force 
is Tmalpv^. Equating the two forces, we have 

Ap/l = 2pv m */a. (43.2) 

Equations (43.1) and (43.2) determine, through the parameter v m , the relation 
between the velocity of flow in the pipe and the pressure gradient. This 
relation is called the resistance law of the pipe. Expressing v m in terms of 
Ap// by (43.2), and substituting in (43.1), we obtain the resistance law in the 
form 

U = V(a&Pl2b*pl) log[(a/vW(a&pl2 P l)]. (43.3) 

In this formula it is customary to introduce what is called the resistance 
coefficient of the pipe, a dimensionless quantity defined as 

laLpll 
A = — . (43.4) 

The dependence of A on the dimensionless Reynolds number R = 2aU/v is 
given in implicit form by the equation 

1/VA = 0-85 log(RVA)-0-55. (43.5) 

We have here substituted for b the value (42.2) and added to the logarithm 
an empirically determined constant.^ The resistance coefficient determined 
by this formula is a slowly decreasing function of the Reynolds number. 
For comparison, we give the resistance law for laminar flow in a pipe. Intro- 
ducing the resistance coefficient in formula (17.10), we obtain 

A = 64/R. (43.6) 

In laminar flow the resistance coefficient diminishes with increasing Reynolds 
number more rapidly than in turbulent flow. 

Fig. 22 shows a logarithmic graph of A as a function of R. The steep 
straight line corresponds to laminar flow (formula (43.6)), and the less 
steep curve (which is almost a straight line also) to turbulent flow. The 
transition from the first line to the second occurs, as the Reynolds number 
increases, at the point where the flow becomes turbulent; this may occur 
for various Reynolds numbers, depending on the actual conditions (the 
intensity of the perturbations ; see §29). The resistance coefficient increases 
abruptly at the transition point. 



t The coefficient of the logarithm in this formula is given to correspond with that in formula 
(42.8) for the logarithmic velocity profile. Only in this case does formula (43.5) have the theoretical 
significance of being a limiting formula for turbulent flow at sufficiently large values of the Reynolds 
number. If the values of the two constants appearing in formula (43.5) are chosen arbitrarily, it can 
only be a purely empirical formula for the dependence of A on R. In that case, however, there would 
be no reason to prefer it to any other simpler empirical formula which adequately represents the 
experimental results. 



§43 



Turbulent flow in pipes 



165 



So far we have assumed that the wall surface is fairly smooth. If it is 
rough, the formulae obtained above may be somewhat changed. As a measure 
of the roughness of the wall, we can take the order of magnitude of the 
projections, which we shall denote by d. The relative magnitudes of d 
and the thickness yo of the sublayer are of importance. If yo is large compared 
with d, the roughness is unimportant; this is what is meant by saying that 
the surface is fairly smooth. If yo and d are of the same order of magnitude, 
no general formulae can be obtained. 




In the opposite limiting case of extreme roughness (d > yo), some general 
relations can again be established. In this case we clearly cannot speak of a 
viscous sublayer. Turbulent flow occurs around the projections from the 
surface, and this flow is characterised by the quantities p,cr,d; the viscosity v, 
as usual, cannot appear directly. The velocity of this flow is of the order of 
magnitude of v#, the only quantity at our disposal having the dimensions of 
velocity. Thus we see that, in flow along a rough surface, the velocity 
becomes small (~ v#) at distances y ~ d, instead of v ~ yo as for flow 
along a smooth surface. Hence it is clear that the velocity distribution is given 
by a formula which is obtained from (42.7) by substituting d for v/v*. Thus 

u = (v*lb)log(yld). (43.7) 

The formulae for flow in a pipe must be changed similarly. It is sufficient 
simply to replace vjv m in them by d. For the resistance law we have, instead of 
(43.3), the formula 

U = ^{akpjltfpl) log(a/d). (43.8) 

The argument of the logarithm is now a constant, and does not involve the 
pressure gradient as (43.3) did. We see that the mean velocity is now simply 
proportional to the square root of the pressure gradient in the pipe. If we 
introduce the resistance coefficient, (43.8) becomes 

A = 8&2/log2(a/i) = 14/log2(a/</), (43.9) 

i.e. A is a constant and does not depend on the Reynolds number. 



166 Boundary Layers §44 

§44. The turbulent boundary layer 

The fact that we have obtained a logarithmic velocity distribution which 
formally holds in all space for plane-parallel turbulent flow is due to our 
having considered flow along a surface of infinite area. In flow along the 
surface of a finite body, only the motion at short distances from the surface — 
in the boundary layer — has a logarithmic profile.f We may mention also that 
a turbulent boundary layer can exist both under a fluid moving turbulently 
in the main stream and under a laminar flow. 

The decrease in the mean velocity, both in the turbulent and in the 
laminar boundary layer, is due ultimately to the viscosity of the fluid. The 
effect of the viscosity appears in the turbulent boundary layer in a rather 
unusual manner, however. The manner of variation of the mean velocity in 
the layer does not itself depend directly on the viscosity; the viscosity appears 
in the expression for the velocity gradient only in the viscous sublayer. The 
total thickness of the boundary layer, however, is determined by the viscosity, 
and vanishes when the viscosity is zero (see below). If the viscosity were 
exactly zero, there would be no boundary layer. 

Let us apply the results of §43 to a turbulent boundary layer formed in 
flow along a thin flat plate, such as was discussed in §39 with respect to 
laminar flow. At the boundary of the turbulent layer, the fluid velocity is 
almost equal to the velocity of the main stream, which we denote by U. 
To determine this velocity at the boundary we can, however, use formula 
(42.7) with logarithmic accuracy, putting the thickness 8 of the boundary 
layer instead of y. Equating the two expressions, we obtain 

U = (v*/b)log(v*8/v). (44.1) 

Here U is a constant parameter for a given flow; the thickness 8, however, 
varies along the plate, and v m is therefore also a slowly varying function of x. 
Formula (44.1) is inadequate to determine these functions; we need some 
other equation, relating v% and 8 to x. 

To obtain this, we use the same arguments as in deriving formula (36.3) 
for the width of the turbulent wake. As there, the derivative d8/dx must 
be of the order of the ratio of the velocity along the jy-axis to that along 
the ar-axis at the boundary of the layer. The latter velocity is of the order 
of U, while the former is due to the fluctuating velocity, and is therefore 
of the order of v m . Thus dSjdx ~ v%/U, whence 

8 ~ v*xjU. (44.2) 



t The thickness of the boundary layer increases along the surface of the body in the direction 
of flow, according to a law which we shall determine below. This explains why, for flow in a pipe, 
the logarithmic profile holds for the whole cross-section of the pipe. The thickness of the boundary 
layer at the wall of the pipe increases away from the point of entry of the fluid. At some finite distance 
from this point, the boundary layer fills almost the whole cross-section of the pipe. Hence, if we 
suppose the pipe sufficiently long and ignore its inlet section, the flow in the whole pipe will be of 
the same kind as in the turbulent boundary layer. We may recall that a similar situation occurs for 
laminar flow in a pipe. Such a flow obeys Poiseuille's formula for all Reynolds numbers. In Poiseuille 
flow the viscosity is important at all distances from the walls, and its effect is never limited to a thin 
layer adjoining them. 



§44 The turbulent boundary layer 167 

Formula (44.1) and (44.2) together determine v m and 8 as functions of the 
distance x.f These functions, however, cannot be written explicitly. We 
shall express 8 in terms of an auxiliary quantity. Since v # is a slowly varying 
function of x, it is seen from (44.2) that the thickness of the layer varies 
essentially as x. We may recall that the thickness of the laminar boundary 
layer increases as \/x y i.e. more slowly than that of the turbulent boundary 
layer. 

Let us determine the dependence on x of the frictional force cr acting 
on unit area of the plate. This dependence is given by two formulae : 

a = P v* 2 , U = {v m jb) Xogip^xjUv). 
The latter is obtained by substituting (44.2) in (44.1), and is valid to logarith- 
mic accuracy. We introduce a drag coefficient c (referred to unit area of the 
plate), defined as the dimensionless ratio 

c = 2aj P W = 2{vJU) 2 . (44.3) 

Then, eliminating v # from the two equations given, we obtain the following 
equation, which gives (to logarithmic accuracy) c as an implicit function of x : 

V(2#7<0 = log(cRs), R x = Ux/v. (44.4) 

To increase the accuracy of this formula, we may add an empirical numerical 
constant to the logarithm. Such a formula is, 

l/y/c = 1-7 log(cR x ) + 3-0. (44.5) 

The drag coefficient c given by this formula is a slowly decreasing function 
of the distance x. 

Finally, let us express the thickness of the boundary layer in terms of the 
function c(x). We have v m = \/( a lp) = UVfa)- Substituting in (44.2), we 
find 

8 = constant x x\/c. (44.6) 

This formula may be written with the equality sign, of course, only in cases 
of a turbulent boundary layer under a laminar flow, when 8 has an exact 
significance (the turbulent region being, as always, sharply distinct from 
the laminar region). The constant factor in (44.6) has to be determined from 
experimental results. 

PROBLEMS 

Problem 1. Determine from formula (44.5) the total force acting on the two sides of the 
plate. 

Solution. The required force per unit length of the edge of the plate is 

I 



= 2 f or dx, 




t If there is a laminar boundary layer of considerable extent on the plate, then x must, strictly 
speaking, be reckoned as approximately the distance from the point where the laminar layer becomes 
turbulent. 



168 Boundary Layers §45 

where / is the length of the plate. Introducing in place of F the drag coefficient 

C = F/ipW.21, 



we find 



1 r 
C = - cdx. 

*■ o 



If we take only terms containing the logarithm to the highest (first) power, then the above 
integral is simply c(Z), the value of c for x = I. In order to obtain a more exact value for C, 
corresponding to formula (44.5), we must effect the integration taking account of terms of 
the next order, which contain the logarithm to the zero power. To do so, we write 

f , I f dc 

\ cdx = [#c] — x — dx. 
o o "* 

The derivative dcfdx is calculated by means of formula (44.5), which we write in the form 
c = If A 2 log 2 Bxc, obtaining to the necessary accuracy 

2 r 2 1 
C = c(l) + = c(l)\l + , 

and so 

V V 5 nA \og{BlCje). 

Substituting the values of A and B from (44.5), we obtain the following formula, which gives 
the total drag coefficient C as a function of the Reynolds number R = Ulfv: 

1/VC= l-71og(CR)+l-3. 

For large R, the drag coefficient given by this formula decreases as 1/log 2 R. For the laminar 
boundary layer, C decreases as 1/VR (see (39.16)), i.e. more rapidly. Thus we can say that, 
for large Reynolds numbers, the frictional force in a turbulent boundary layer is greater than 
in a laminar one. 

Problem 2. Determine the drag coefficient of a rough plate as a function of the Reynolds 
number, for a turbulent boundary layer. 

Solution. Substituting in place of the thickness y (^ vjv*) of the laminar sublayer the 
dimension d of the projections, we obtain from (44.1) and (44.2) U = (vjb) log(xv JUd). 
Introducing the drag coefficient c, we hence have 0-59J\/c = log(x\/cld). Similarly, the 
total drag coefficient for the plate is (again to logarithmic accuracy) 0-59/t/C = log(l\/Cfd). 
We may point out that the drag coefficient for a rough plate is independent of the Reynolds 
number. 

§45. The drag crisis 

From the results obtained in the previous sections we can draw important 
conclusions concerning the law of drag for large Reynolds numbers, i.e. the 
relation between the drag force acting on the body and the value of R when 
the latter is large. 



§45 The drag crisis 169 

The flow pattern for large R (the only case we shall discuss) has already 
been described, and is as follows. Throughout the main body of the fluid 
(i.e. everywhere except in the boundary layer, which does not here concern 
us) the fluid may be regarded as ideal, with potential flow everywhere 
except in the turbulent wake. The width of the wake depends on the position 
of the line of separation on the surface of the body. It is important to note 
that, although this position is determined by the properties of the boundary 
layer, it is found to be independent of the Reynolds number, as we have seen 
in §40. Thus we can say that the whole flow pattern for large Reynolds 
numbers is almost independent of the viscosity, i.e. of R (so long as the boun- 
dary layer remains laminar; see below). 

Hence it follows that the drag also must be independent of the viscosity. 
There remain at our disposal only three quantities: the velocity U of the 
main stream, the fluid density p and the dimension / of the body. From these 
we can construct only one quantity having the dimensions of force, namely 
pTJH 2 . Instead of the squared linear dimension of the body I 2 , we introduce, 
as is customarily done, the proportional quantity S, the area of a cross-section 
transverse to the direction of flow, putting 

F = constant x P U 2 S, (45.1) 

where the constant is a number depending only on the shape of the body. 
Thus the drag must be (for large R) proportional to the cross-sectional area 
of the body and to the square of the main-stream velocity. We may recall 
for comparison that, for very small R (^ 1), the drag is proportional to the 
linear dimension of the body and to the velocity itself (F ~ vplU; see §20).f 
It is customary, as we have said, to introduce, in place of the drag force 
F, the drag coefficient C defined by C = FJlpU 2 S. This is a dimensionless 
quantity, and can depend only on R. Formula (45.1) becomes 

C = constant, (45.2) 

i.e. the drag coefficient depends only on the shape of the body. 

The above behaviour of the drag force cannot continue to arbitrarily 
large Reynolds numbers. The reason is that, for sufficiently large R, the 
laminar boundary layer (on the surface of the body as far as the line of separa- 
tion) becomes unstable and hence turbulent. However, the whole boundary 
layer does not become turbulent, but only some part of it. The surface of the 
body may therefore be divided into three parts : at the front there is a laminar 
boundary layer, then a turbulent layer, and finally the region beyond the 
line of separation. 

The onset of turbulence in the boundary layer has an important effect 
on the whole pattern of flow in the main stream. It leads to a considerable 
displacement of the line of separation towards the rear of the body (i.e. 



t The flow past a bubble of gas is a special case, where the drag remains proportional to U even 
for large R; see Problem. 



170 Boundary Layers §45 

downstream), so that the turbulent wake beyond the body is contracted, as 
shown in Fig. 23, where the wake region is shaded.f The contraction of the 
turbulent wake leads to a reduction of the drag force. Thus the onset of 
turbulence in the boundary layer at large Reynolds numbers is accompanied 
by a decrease in the drag coefficient, which falls off by a considerable factor 
over a relatively narrow range of Reynolds numbers near 10 5 . We shall call 
this phenomenon the drag crisis. The decrease in the drag coefficient is so great 
that the drag itself, which for constant C is proportional to the square of the 
velocity, actually diminishes with increasing velocity in this range of Reynolds 
numbers. 





Fig. 23 



It may be mentioned that the degree of turbulence in the main stream 
affects the drag crisis; the greater the incident turbulence, the sooner the 
boundary layer becomes turbulent (i.e. the smaller is R when this happens). 
The decrease in the drag coefficient therefore begins at a smaller Reynolds 
number, and extends over a wider range of R. 

Figs. 24 and 25 give experimentally obtained graphs showing the drag 
coefficient as a function of the Reynolds number R = Udjv for a sphere; 
Fig. 24 is plotted logarithmically. For very small R ( <^ 1), the drag coefficient 
decreases according to C = 24/R (Stokes' formula). The decrease in C 
continues more slowly as far as R « 5 x 10 3 , where C reaches a minimum, 
beyond which it increases somewhat. In the range of Reynolds numbers 
2 x 10 4 to 2 x 10 5 , the law (45.2) holds, i.e. C is almost constant. The drag 
crisis occurs for R between 2 x 10 5 and 3 x 10 5 , and the drag coefficient 
diminishes by a factor of 4 or 5. 

For comparison, we may give an example of flow in which there is no 
critical Reynolds number. Let us consider flow past a flat disk in the direction 
perpendicular to its plane. In this case the location of the separation is obvious 
from purely geometrical considerations: it is clear that separation occurs at 
the edge of the disk and does not move from there. Hence, as R increases, the 

f For example, in transverse flow past a long cylinder, the onset of turbulence in the boundary 
layer moves the point of separation from 95° to 60° (where the azimuthal angle on the cylinder is 
measured from the direction of flow). 



§45 



The drag crisis 



171 



drag coefficient of the disk remains constant, and there is no drag 
crisis. 

It must be borne in mind that, for the high velocities at which the 
drag crisis occurs, the compressibility of the fluid may begin to have 
a noticeable effect. The parameter which characterises the extent of this 
effect is the Mack number M = U/c, where c is the velocity of sound; if 
M < 1, the fluid may be regarded as incompressible (§10). Since, of the 
two numbers M and R, only one contains the dimension of the body, these 
two numbers can vary independently. 



100 




V 






































- 


60 




> 


s 








































\ 






































20 












































8 
























































































C A 
























































































2 
1-5 












































1-0 
























































































0*6 
























































































0"3 






















































































C 


>1 




1 





2 


51 







1 


D* 




1 


3 3 




1 


O 4 




r 


D b 




K 



R 
Fig. 24 




10 5 2i0 5 3-10 5 4-10 5 5-10 5 
R 
Fig. 25 



The experimental data indicate that the compressibility has in general 
a stabilising effect on the flow in the laminar boundary layer. When M 



172 Boundary Layers §46 

increases, the critical value of R increases. For example, when M for a 
sphere changes from 0-3 to 0-7, the drag crisis is postponed from 
R« 4xl0 5 toR« 8x105. 

We may also mention that, when M increases, the position of the point 
of separation in the laminar boundary layer moves upstream, towards the 
front of the body, and this must lead to some increase in the drag. 

PROBLEM 

Determine the drag force on a gas bubble moving in a liquid at large Reynolds numbers 
(V. G. Levich 1949). 

Solution. At the boundary between the liquid and the gas the tangential fluid velocity 
component does not vanish, but its normal derivative does (we neglect the viscosity of the 
gas). Hence the velocity gradient near the boundary will not be particularly high, and there 
will be no boundary layer in the sense of §39 ; there will therefore be no separation over almost 
the whole surface of the bubble. In calculating the energy dissipation from the volume integral 
(16.3) we can therefore use in all space the velocity distribution corresponding to potential 
flow past a sphere (§10, Problem 2), neglecting the surface layer of liquid and the very narrow 
turbulent wake. Using the formula obtained in §16, Problem, we find 

£kin = —q (7— ) 27r# 2 sin0d0 = -\2irqRU 2 . 
Hence we see that the required dissipative drag isf -F = \2tttjRU. 

§46. Flow past streamlined bodies 

The question may be asked what should be the shape of a body (of a given 
cross-sectional area, say) for the drag on it resulting from motion in a fluid 
to be as small as possible. It is clear from the above that, for this to be so, 
the separation must be as far back as possible: the separation must occur 
near the rear end of the body, so that the turbulent wake is as narrow as 
possible. We know already that the appearance of separation is facilitated by 
the presence of a rapid downstream increase in the pressure along the body. 
Hence the body must have a shape such that the variation in pressure along 
it, where the pressure is increasing, takes place as slowly and smoothly as 
possible. This can be achieved by giving the body a shape elongated in the 
direction of flow, tapering smoothly to a point downstream, so that the flows 
along the two sides of the body meet smoothly without having to go round any 
corners or turn through a considerable angle from the direction of the 
main stream. At the front end the body must be rounded ; if there were an 
angle here, the fluid velocity at its vertex would become infinite (see §10, 
Problem 6), and consequently the pressure would increase rapidly down- 
stream, with separation inevitably resulting. 

All these requirements are closely satisfied by shapes of the kind shown 
in Fig. 26. The profile shown in Fig. 26b may be, for example, the cross- 
section of an elongated solid of revolution, or the cross-section of a body with 



f The range of applicability of this formula is actually not large, since, when the velocity increases 
sufficiently, the bubble ceases to be spherical. 



§46 Flow past streamlined bodies 173 

a large "span" (we conventionally call such a body a wing). The cross- 
sectional profile of a wing may be unsymmetrical, as in Fig. 26a. In flow 
past a body of this shape, separation occurs only in the immediate neighbour- 
hood of the pointed end, and consequently the drag coefficient is relatively 
small. Such bodies are said to be streamlined. 



Fig. 26 



The direct friction of the fluid on the surface in the boundary layer is 
important in the drag on streamlined bodies. This effect for non-streamlined 
bodies (which were considered in the previous section) is relatively small and 
therefore, in practice, of no significance. In the opposite limiting case of 
flow parallel to a flat disk, the effect becomes the only source of drag (§39). 

In flow past a streamlined wing inclined to the main stream at a small 
angle a, called the angle of attack (Fig. 26), a large lift force F y is developed, 
while the drag F x remains small, and the ratio F y \F x may therefore reach 
large values (~ 10-100). This continues, however, only while the angle 
of attack is small ( < 10°). For larger angles the drag rises very rapidly, and 
the lift decreases. This is explained by the fact that, at large angles of attack, 
the body ceases to be streamlined : the point of separation moves a considerable 
way towards the front of the body, and the wake consequently becomes 
wider. It must be borne in mind that the limiting case of a very thin body, i.e. 
a flat plate, is streamlined only for a very small angle of attack; separation 
occurs at the leading edge of the plate when it is inclined at even a small 
angle to the main stream. 

The angle of attack a is, by definition, measured from the position of the 
wing for which the lift force is zero. For small angles of attack, we can 
expand the lift as a series of powers of a. Taking only the first term, we can 
suppose that the force F y is proportional to a. Next, by the same dimensional 
arguments as for the drag force, the lift must be proportional to pU 2 . Intro- 
ducing also the span l z of the wing, we can write 

F y = constant x p U 2 cd x l z , (46.1) 

where the numerical constant depends only on the shape of the wing and not, 
in particular, on the angle of attack. For very long wings, the lift may be 



174 Boundary Layers §46 

supposed proportional to the span, in which case the constant depends only 
on the shape of the cross-section of the wing. 

Instead of the lift on the wing, the lift coefficient is often used; it is defined 
as 

Cy = FylipU%l z . (46.2) 

For very long wings, according to what was said above, the lift coefficient is 
proportional to the angle of attack, and depends on neither the velocity nor 
the span : 

Cy = constant x a. (46.3) 

To calculate the lift on a streamlined wing by means of Zhukovskii's 
formula, it is necessary to determine the velocity circulation Y. This is 
done as follows. We have potential flow everywhere outside the wake. In 
the present case, the wake is very thin, and occupies on the surface of the 
wing only a very small area near its pointed trailing edge. Hence, to determine 
the velocity distribution (and therefore the circulation T), we can solve the 
problem of potential flow of an ideal fluid round a wing. The existence of the 
wake is taken into account by the presence of a tangential discontinuity, 
extending into the fluid from the sharp trailing edge of the wing, where the 
potential has a discontinuity <j> 2 — <j>\ = Y. As has been shown in §37, the 
derivative d<f>fdz also has a discontinuity on this surface, while the derivatives 
d<f>jdx and d<f>[dy are continuous. For a wing of finite span, the problem in 
this form has a unique solution. The finding of the exact solution is very 
complicated, however. The problem has been solved by N. E. KocHiNf 
for a wing in the form of a circular disk inclined at a small angle of attack. 

If the wing is very long (and has a uniform cross-section), then, regarding 
it as infinite in the ^-direction, we may regard the flow as two-dimensional 
(in the xy-plane). It is evident from symmetry that the velocity v z = d<f>jdz 
along the wing must be zero. In this case, therefore, we must seek a solution 
in which only the potential has a discontinuity, its derivatives being con- 
tinuous; in other words, there is no surface of tangential discontinuity, 
and we have simply a many-valued function <f>(x, y), which receives a finite 
increment Y when we go round a closed contour enclosing the profile of the 
wing. In this form, however, the problem of two-dimensional flow has no 
unique solution, since it admits solutions for any given discontinuity of the 
potential. To obtain a unique result, we must require the fulfilment of 
another condition, first formulated by S. A. Chaplygin in 1909. 

This condition, called the Zhukovskii-Chaplygin condition, consists in 
requiring that the fluid velocity does not become infinite at the sharp trailing 
edge of the wing; in this connection we may recall that, when an ideal fluid 
flows round an angle, the fluid velocity in general becomes infinite, according 
to a power law, at the vertex of the angle (§10, Problem 6). We can say that 



t Prikladnaya matematika i mekhamka 4, 3, 1940; 9, 13, 1945. 



§47 Induced drag 175 

the condition stated implies that the jets coming from the two sides of the 
wing must meet smoothly without turning through an angle. When this 
condition is fulfilled, of course, the solution of the problem of potential flow 
gives a pattern very like the true one, where the velocity is everywhere finite 
and separation occurs only at the trailing edge. The solution now becomes 
unique and, in particular, the circulation T needed to calculate the lift force 
has a definite value. 

§47. Induced drag 

An important part of the drag on a streamlined wing (of finite span) is 
formed by the drag due to the dissipation of energy in the thin turbulent wake. 
This is called the induced drag. 

It has been shown in §21 how we may calculate the drag force due to the 
wake by considering the flow far from the body. Formula (21.1), however, 
is not applicable in the present case. According to that formula, the drag is 
given by the integral of v x over the cross-section of the wake, i.e. the discharge 
through the wake. On account of the thinness of the wake beyond a stream- 
lined wing, however, the discharge is small in the present case, and may be 
neglected in the approximation used below. 

As in §21, we write the force F x as the difference between the total fluxes 
of the ^-component of momentum through the planes x = x\ and x = #2 
passing respectively far behind and far in front of the body. Writing the 
three velocity components as U+v x , v y , v z , we have for the component Yi xx 
of the momentum flux density the expression U xx = p + p(U+v x ) 2 , so that 
the drag force is 

F * = ( 1/ " 17 )[P + p(U+Vx) 2 ]dydz. (47.1) 

x=x t x=x x 

On account of the thinness of the wake, we can neglect, in the integral over 
the plane x = x±, the integral over the cross-section of the wake, and so 
integrate only over the region outside the wake. In that region, however, 
we have potential flow, and Bernoulli's equation /> + |p(U+v) 2 = p Q + % p U 2 
holds, whence 

p = po-pUv x -%p(v x 2 +v y z+v z 2 ). (47.2) 

Here we cannot neglect the quadratic terms as we did in §21, since it is these 
terms which determine the required drag force in the case under considera- 
tion. Substituting (47.2) in (47.1), we obtain 

F * = ( // ~ // )lPo+pU 2 +pUv x +± P (v x Z- V-«fc 2 )]dyd*. 
x=x t x=x t 

The difference of the integrals of the constant po + pU 2 is zero; the difference 



176 Boundary Layers §47 

of the integrals of pUv x is likewise zero, since the mass fluxes 

jjpv x dydz 

through the front and back planes must be the same (we neglect the discharge 
through the wake in the approximation here considered). Next, if we take 
the plane x = x% sufficiently far in front of the body, the velocity v on this 
plane is very small, so that the integral of \p{v x 2 — v y 2 — v z 2 ) over this plane 
may be neglected. Finally, in flow past a streamlined wing, the velocity v x 
outside the wake is small compared with v y and v z . Hence we can neglect v x 2 
compared with v y 2 + v z 2 in the integral over the plane x = x\. Thus we obtain 

Fx = h jj (v y 2 +v z 2 )dydz, (47.3) 

where the integration is over a plane x = constant lying at a great distance 
behind the body, the cross-section of the wake being excluded from the region 
of integration.-f 

The drag on a streamlined wing calculated in this way can be expressed in 
terms of the velocity circulation Y which determines the lift also. To do 
this, we first of all notice that, at sufficiently great distances from the body, 
the velocity depends only slightly on the co-ordinate x, and so we can regard 
v v{y> %) an( l v z(y, %) as the velocity of a two-dimensional flow, supposed 
independent of x. It is convenient to use as an auxiliary quantity the stream 
function (§10), so that v z = difjjdy, v y = -dxjsjdz. Then 



->!![{$)'*(& 



dydz, 



where the integration over the vertical co-ordinate y is from + oo to ji 
and from y% to — oo, where y\ and y^ are the co-ordinates of the upper and 
lower boundaries of the wake (see Fig. 18, §37). Since we have potential flow 
(curlv = 0) outside the wake, d 2 ip{dy 2 + d 2 ^jdz 2 = 0. Using the two- 
dimensional Green's formula, we thus find 



F x = -lp§<fj{dil>{dn)dl, 



where the integral is taken along a contour bounding the region of integration 
in the original integral, and djdn denotes differentiation in the direction of the 
outward normal to the contour. At infinity ift = 0, and so the integral is taken 



t To avoid misunderstanding we should point out the following. Formula (47.3) may give the im- 
pression that the velocities v y , v z do not decrease in order of magnitude as x increases. This is true 
so long as the thickness of the wake is small compared with its width, as we have assumed in deriving 
formula (47.3). At very large distances behind the wing, the wake finally becomes so thick that it 
becomes approximately circular in cross-section. At this point, formula (47.3) is invalid, and %, 
v z diminish rapidly with increasing x. 



§47 Induced drag 177 

round the cross-section of the wake by the yz-ptene, giving 

Here the integration is over the width of the wake, and the difference in the 
brackets is the discontinuity of the derivative 8iff/8y across the wake. Since 

8ifsf8y = v z = 8<f)l8z, we have 

\8y) 2 Uj/i \8z/ 2 \8zJ1 dz' 
so that 

F* = ipJ#dr/d*)d*. 

Finally, we use a formula from potential theory, 

*--=J[(S).-(S)J— * 

where the integration is along a plane contour, r is the distance from dl 
to the point where iff is to be found, and the expression in brackets is the 
given discontinuity of the derivative of iff in the direction normal to the 
contour.-)- In our case the contour of integration is a segment of the #-axis, 
so that we can write the value of the function ip(y y z) on the sr-axis as 

**>4[(E-(i),H-''"' 

= _IW log |,_^<. 

2ttJ dz' Si ' 

Finally, substituting this in F x , we obtain the following formula for the 
induced drag: 



4tJJ 



' 'dl» dr(ar') , , /1J A , , AnA ^ 

— — — — log \z-z'\dzdz' (47 A) 

. . dz dz' 




(L. Prandtl, 1918). The span of the wing is here denoted by l z = /, and 
the origin of z is at one end of the wing. 

If all the dimensions in the ^-direction are increased by some factor (r 



t This formula gives, in two-dimensional potential theory, the potential due to a charged plane 
contour with a charge density 

[(8iffl8n)2-(diPldn) 1 ]/2'jT. 



178 Boundary Layers §47 

remaining constant), the integral (47.4) remains constant, f This shows that 
the total induced drag on the wing remains of the same order of magnitude 
when its span is increased. In other words, the induced drag per unit length 
of the wing decreases with increasing length. $ Unlike the drag, the total 
lift force 

F y = - P U JFdz (47.5) 

increases almost linearly with the span of the wing, and the lift per unit length 
is constant. 

The following method is convenient for the actual calculation of the inte- 
grals (47.4) and (47.5). Instead of the co-ordinate z, we introduce a new 
variable 0, defined by 

* = |/(l-cos0) (0 < 6 ^ tt). (47.6) 

The distribution of the velocity circulation is written as a Fourier series: 



T = -2UI £ A n smnd. (47.7) 

The condition that T = at the ends of the wing (z = and /, or 6 = 
and it) is then fulfilled. 

Substituting the expression (47.7) in (47.5) and effecting the integration 
(using the orthogonality of the functions sin and sin n6 for n ^ 1), we 
obtain F y = \pU 2 -nl 2 A\. Thus the lift force depends only on the first 
coefficient in the expansion (47.7). For the lift coefficient (46.2) we have 

C y = ttA^i, (47.8) 

where we have introduced the ratio A = l\l x of span to width of the wing. 
To calculate the drag, we rewrite formula (47.4), integrating once by parts : 

p } r , s dlY*') dz'dz 
o o 



t To avoid misunderstanding, we should mention that it does not matter that the logarithm in 
the integrand is increased by a constant when the unit of length is changed. For the integral which 
differs from that in (47.4) by having a constant instead of log |«— «'| is zero, since 



j (dr/dz)dz = r, 



and the definite integral is zero because T vanishes at the edges of the wake. 

% In the limit of infinite span, the induced drag per unit length is zero. In reality, a small amount 
of drag remains, determined by the discharge through the wake (i.e. the integral //»* dy dz), which 
we have neglected in deriving formula (47.3). This drag includes both the frictional drag and the 
remaining part due to dissipation in the wake. 



§48 The lift of a thin wing 179 

It is easily seen that the integral over z' must be taken as a principal value. 
An elementary calculation, with the substitution (47.7),f leads to the following 
formula for the induced drag coefficient : 



C x = ttX J^nA n 2. (47.10) 

The drag coefficient for a wing is defined as 

C x = F x l\ P Un x l z , (47.11) 

being referred, like the lift coefficient, to unit area in the xsr-plane. 

PROBLEM 

Determine the least value of the induced drag for a given lift and a given span l z — I. 
Solution. It is clear from formulae (47.8) and (47.10) that the least value of C x for given 
C y (i.e. for given Ay) is obtained if all A n for n ^ 1 are zero. Then 

C^min = Cy^jirX. (1) 

The distribution of velocity circulation over the span is given by the formula 

T= -~Ul x C y ^[z{l-z)]. (2) 

TTL 

If the span is sufficiently large, then the flow round any cross-section of the wing is approxi- 
mately two-dimensional flow round a wing of infinite length and the same cross-section. 
In this case we can say that the circulation distribution (2) is obtained for a wing whose shape 
in the ara-plane is an ellipse with semi-axes \l% and \l. 

§48. The lift of a thin wing 

The problem of calculating the lift force on a wing amounts, by Zhukovskii's 
theorem, to that of finding the velocity circulation I\ A general solution of 
the latter problem can be given for a thin streamlined wing of infinite span, 
the cross-section being the same at every point. { The elegant method of 



t In integrating over z' we need the integral 



cos«0' 7rsin«0 

dd' = 



'f - • 

J cos 6' — cos 6 sin 8 





In integrating over z we use the fact that 

sin nd sin md d# = \tt (m = n), 

o 

= (m ^ n). 

J A more detailed account of the theory of two-dimensional incompressible flow past a wing is 
given by N. E. Kochin, I. A. Kibel* and N. V. Roze, Theoretical Hydromechanics {Teoreticheskaya 
gidromekhanika), Part 1, 4th ed., Moscow 1948; L. I. Sedov, Two-dimensional Problems of Hydro- 
dynamics and Aerodynamics (Ploskie zadachi gidrodinamiki i aerodinamiki), Moscow 1950. 



180 



Boundary Layers 



§48 



solution given below is due to M. V. Keldysh and L. I. Sedov (1939). 

Let y = £i(x) and y = ^(x) be the equations of the lower and upper parts 
of the cross-sectional profile (Fig. 27). We suppose this profile to be thin, only 
slightly curved, and inclined at a small angle of attack to the main stream 
(the #-axis); that is, both £i, £2 themselves and their derivatives £1', £2' 
are small, i.e. the normal to the profile contour is everywhere almost parallel 




Fig. 27 



to the j-axis. Under these conditions, we may suppose the perturbation v 
in the fluid velocity, caused by the presence of the wing, to be everywheref 
small compared with the main-stream velocity U. The boundary condition 
at the surface of the wing is v y /U = £' for y = £. By virtue of the assump- 
tions made, we can suppose this condition to hold for y = 0, and not for 
y — £. Then we must have on the axis of abscissae between x = and 
x == i x == a 

v y = Ut,2\x) for y -»0 + , v y = U&(x) for y ->0-. (48.1) 

In order to apply the methods of the theory of functions of a complex 
variable, we introduce the complex velocity dwfdz = v x — iv y (cf. §10), 
which is an analytic function of the variable z = x+iy. In the present case 
this function must satisfy the conditions 

im(dw/dz) = — U£,2'(x) for y ~+ + , 

im(d«;/d*) = - Z7£i'(«) for y ->0-, ^^ 

on the segment (0, a) of the axis of abscissae. 

To solve the above problem, we first represent the required velocity 
distribution v(x,y) as a sum v = v + + v~ of two distributions having the 
following symmetry properties: 

©-*(*, ~y) = v- x {x,y), v- y (x, -y) = -v~ y {x y y), 

v + x (x, -y) = -v+ x (x,y) } v+ y (x, -y) = v+ y (x,y). 

These properties of the separate distributions v~ and v + do not violate the 
equation of continuity or that of potential flow, and, since the problem is 
linear, the two distributions may be sought separately. 



t Except in a small region near the rounded leading edge of the wing. 



§48 



The lift of a thin wing 



181 



(48.4) 



The complex velocity is correspondingly represented as a sum 

to' = zo'+ + w'_, 

and the boundary conditions on the segment (0, a) for the two terms of the 
sum are 

[imw'+]y^ 0+ = [imw'+] y -+o- = -W(£i+&)> 
[im w'_] y ^o + = - [im w'_] y ^o_ = \ U(U' - £ 2 ')- 
The function zo'_ can be determined at once by Cauchy's formula: 

w -{*) = -^-.ty-, d £ 

Z7rt J <; — z 

L 

where the integration in the plane of the complex variable £ is along a circle 
L of small radius centred at the point £ = z (Fig. 28). The contour L can 




Fig. 28 



be replaced by a circle C" of infinite radius and a contour C traversed clock- 
wise ; the latter can be deformed into the segment (0, a) twice over. The 
integral along C is zero, since w'(z) vanishes at infinity. The integral 
along C gives 



to _ = 



U r&'(0-£i'(0 



! 



£-* 



<l|. 



(48.5) 



Here we have used the boundary values (48.4) of the imaginary part of «/_ 
on the segment (0, a), and the fact that, by the symmetry conditions (48.3), 
the real part of «/_ is continuous across this segment. 

To find the function «?' + , we have to apply Cauchy's formula, not to this 
function itself, but to the product w' + (z)g(z), where g(z) = s/[zj{z— a)]> 
and the square root is taken with the plus sign for z — x > a. On the 
segment (0, a) of the real axis, the function g(z) is purely imaginary and dis- 
continuous :g(x+i0) = — g(x— iO) = — i^[xj{a — x)]. It is clear from these 
properties of the function g(z) that the imaginary part of the product gio'+ 



182 Boundary Layers §48 

is discontinuous across the segment (0, a), while the real part is continuous, 
as with the function w'_. Hence we have, exactly as in the derivation of 
formula (48.5), 

U f &'(fl + & '(fl 
«> +(*)£(*) = -z- 1 g(£+i0)dl 

Lit J S~Z 



Collecting the above expressions, we have the following formula for the 
velocity distribution in flow past a thin wing: 



:d£- 



^ = _ _^_ /fZf f &'(fl +&'(!) /_jr_ 

d# 2irt'V # J £-# N a-£ 

o 

"■f*'<fl-*'®«. (48.6) 








Near the rounded leading edge (i.e. for z -> 0), this expression in general 
becomes infinite, the approximation used above being invalid in this region. 
Near the pointed trailing edge (i.e. for z -> a), the first term in (48.6) is 
finite, but the second term becomes infinite, though only logarithmically, f 
This logarithmic singularity is due to the approximation used, and is removed 
by a more exact treatment; there is no power-law divergence at the trailing 
edge, in accordance with the Zhukovskii-Chaplygin condition. The fulfilment 
of this condition is achieved by an appropriate choice of the function g(z) 
used above. 

Formula (48.6) immediately enables us to determine the velocity circulation 
T round the wing profile. According to the general rule (see §10), T is 
given by the residue of the function w'{z) at its simple pole z = 0. The 
required residue is easily found as the coefficient of \jz in an expansion of 
to\z) in powers of \\z about the point at infinity: dzojdz = rj2Triz+... , 
and r is given by the simple formula 

a 

r= uj(K+&)J^-&. (48.7) 

o 

We may point out that only the sum of the functions £i and £2 appears here. 
The lift force is unchanged if the thin wing is replaced by a bent plate whose 
shape is given by the function |(£i+ £2). 

For example, for a wing in the form of a thin plate of infinite length, 
inclined at a small angle of attack a, we have £1 = £2 = a(a — x), and for- 
mula (48.7) gives F — —ircnaU. The lift coefficient for such a wing is 
C y = -pXJTjyWa = 2™. 

t This divergence disappears if £ x and £ 2 vanish as (a — x)k, k > 1, near the trailing edge, i.e. if the 
point at the trailing edge is a cusp. 



CHAPTER V 

THERMAL CONDUCTION IN FLUIDS 

§49. The general equation of heat transfer 

It has been mentioned at the end of §2 that a complete system of equations 
of fluid dynamics must contain five equations. For a fluid in which processes 
of thermal conduction and internal friction occur, one of these equations is, 
as before, the equation of continuity, and Euler's equations are replaced by 
the Navier-Stokes equations. The fifth equation for an ideal fluid is the 
equation of conservation of entropy (2.6). In a viscous fluid this equation 
does not hold, of course, since irreversible processes of energy dissipation 
occur in it. 

In an ideal fluid the law of conservation of energy is expressed by 
equation (6.1): 

a 

— ($pv 2 + pe) = -diy\fiv(^v 2 + w)]. 
ot 

The expression on the left is the rate of change of the energy in unit volume of 

the fluid, while that on the right is the divergence of the energy flux density. 

In a viscous fluid the law of conservation of energy still holds, of course: 

the change per unit time in the total energy of the fluid in any volume must 

still be equal to the total flux of energy through the surface bounding that 

volume. The energy flux density, however, now has a different form. 

Besides the flux p\(^v 2 + zo) due to the simple transfer of mass by the motion 

of the fluid, there is also a flux due to processes of internal friction. This 

latter flux is given by the vector v»o', with components (;<</{* (see §16). 

There is, moreover, another term that must be included in the energy flux. 

If the temperature of the fluid is not constant throughout its volume, there 

will be, besides the two means of energy transfer indicated above, a transfer of 

heat by what is called thermal conduction. This signifies the direct molecular 

transfer of energy from points where the temperature is high to those where 

it is low. It does not involve macroscopic motion, and occurs even in a fluid 

at rest. 

We denote by q the heat flux density due to thermal conduction. The 

flux q is related to the variation of temperature through the fluid. This 

relation can be written down at once in cases where the temperature gradient 

in the fluid is not large; in phenomena of thermal conduction we are almost 

always concerned with such cases. We can then expand q as a series of powers 

of the temperature gradient, taking only the first terms of the expansion. The 

constant term is evidently zero, since q must vanish when grad T does so. 

Thus we have 

q = -/cgradT. (49.1) 

183 



184 Thermal Conduction in Fluids §49 

The constant k is called the thermal conductivity. It is always positive, as 
we see at once from the fact that the energy flux must be from points at a 
high temperature to those at a low temperature, i.e. q and grad T must be 
in opposite directions. The coefficient k is in general a function of tempera- 
ture and pressure. 

Thus the total energy flux in a fluid when there is viscosity and thermal 
conduction is p\{\v 2 + w)-\»a' - k grad T. Accordingly, the general law 
of conservation of energy is given by the equation 

d 
— (%pv 2 + pe) =» - div[pv(%v 2 + w) - v • a' - k grad T] . (49.2) 

This equation could be taken to complete the system of fluid-mechanical 
equations of a viscous fluid. It is convenient, however, to put it in another 
form by transforming it with the aid of the equations of motion. To do so, we 
calculate the time derivative of the energy in unit volume of fluid, starting 
from the equations of motion. We have 

d dp by de dp 

Substituting for dpjdt from the equation of continuity and for dw\dt from 
the Navier-Stokes equation, we have 

d 

— (|p^ 2 +pe) = — $v 2 di\(pv) — pv»grad^ 2 — v»grad/> + 
dt 

+ vi- — + p— - e div(pv). 
cxjc at 

Using now the thermodynamic relation de = Tds— p dV = Tds+(plp 2 )dp, 
we find 

de ds P dp ds p 

— =T— + — —=T — div(pv). 

dt dt p 2 dt et p 2 v ' 

Substituting this and introducing the heat function w = e+p[p, we obtain 

d 

— (Ipv 2 + pe) — — (%v 2 + w) div(pv) — pv • grad^w 2 — v • grad/) + 

dt 

ds da ilc 

+ pT Wi . 

dt dxjc 

Next, from the thermodynamic relation dw = Tds+dp/p we have 
grad/) = p grad tv — p T grad s. The last term on the right of the above 
equation can be written 

da iv- d dvi di)* 

= —— (vio'iie) - cr'ac- — = div(va')-ff'i 



dxic dxjc dxjc dxjc 



§49 The general equation of heat transfer 1 85 

Substituting these expressions, and adding and subtracting div(/c grad T), 
we obtain 

d 

— (%pv 2 +pe) == — div[pv(%v 2 + w)— v»o' — KgradT] + 

dt 

I ds \ dvi 

+ P T[ — + vgrad* - a' ik — - - div(ic grad T). (49.3) 

\ dt / dxjc 

Comparing this expression for the time derivative of the energy in unit 
volume with (49.2), we have 

/ ds \ dvi 

pTI — + vgrads = a' ilc — - + div(* grad T). (49.4) 

\ 8t I dxjc 

This equation is called the general equation of heat transfer. If there is no 
viscosity or thermal conduction, the right-hand side is zero, and the equation 
of conservation of entropy (2.6) for an ideal fluid is obtained. 

The following interpretation of equation (49.4) should be noticed. The 
expression on the left is just the total time derivative ds/dt of the entropy, 
multiplied by pT. The quantity ds/dt gives the rate of change of the entropy 
of a unit mass of fluid as it moves about in space, and T ds/dt is therefore 
the quantity of heat gained by this unit mass in unit time, so that pT dsjdt 
is the quantity of heat gained per unit volume. We see from (49.4) that 
the amount of heat gained by unit volume of the fluid is therefore 

v'ik dvildxic + div(K grad T). 

The first term here is the energy dissipated into heat by viscosity, and the 
second is the heat conducted into the volume concerned. 

We expand the term a'acdvijdxk in (49.4) by substituting the expression 
(15.3) for o' ik . We have 

dvi dvi / dvi dvjc dvi \ dvi dvi 

a'ik = f] ( 1 I §ik I + LfT^ik— — • 

dxjc dxjc \dxjc 8xi oxi 1 dxjc oxi 

It is easy to verify that the first term may be written as 

dvi dvjc ^ dvi\' 



and the second is 



\ dxjc dxt dxi / 



dvi dvi dvi dvi 

OXlc OXl OXi OX\ 



Thus equation (49.4) becomes 



ds \ I dvi dvjc dvi \ 2 

- + vgrads = div( K grad T)+± v — - + — - - fS i& —- + 
at I \ dxjc oxi oxi I 

+ £(divv) 2 . (49.5) 



186 Thermal Conduction in Fluids §49 

The entropy of the fluid increases as a result of the irreversible processes 
of thermal conduction and internal friction. Here, of course, we mean not the 
entropy of each volume element of fluid separately, but the total entropy of the 
whole fluid, equal to the integral 

J* psdV. 

The change in entropy per unit time is given by the derivative 

d[ j P sdV]Jdt = j [8{ps)/dt]dV. 

Using the equation of continuity and equation (49.5) we have 

8( P s) 8s 8 P 1 

+ s— = -s div(pv)-pvgrads H div(/c grad T) + 



8t 8t 8t ° T 

7) I 8v t 8v k 8vi\* I 

The first two terms on the right together give -div(psv). The volume 
integral of this is transformed into the integral of the entropy flux psv 
over the surface. If we consider an unbounded volume of fluid at rest at 
infinity, the bounding surface can be removed to infinity; the integrand in the 
surface integral is then zero, and so is the integral itself. The integral of 
the third term on the right is transformed as follows: 

Assuming that the fluid temperature tends sufficiently rapidly to a constant 
value at infinity, we can transform the first integral into one over an infinitely 
remote surface, on which grad T = and the integral therefore vanishes. 
The result is 

d f C K(gradT) 2 r r> / 8vi 8v k 8vi\ 2 

+ f— (divv)2dF. (49.6) 

The first term on the right is the rate of increase of entropy owing to thermal 
conduction, and the other two terms give the rate of increase due to internal 
friction. The entropy can only increase, i.e. the sum on the right of (49.6) 
must be positive. In each term, the integrand may be non-zero even if the 
other two integrals vanish. Hence it follows that the second viscosity 
coefficient £ is positive, as well as k and 77, which we already know are positive. 
It has been tacitly assumed in the derivation of formula (49.1) that the 
heat flux depends only on the temperature gradient, and not on the pressure 
gradient. This assumption, which is not evident a priori, can now be justified 



§49 The general equation of heat transfer 187 

as follows. If q contained a term proportional to grad/>, the expression 
(49.6) for the rate of change of entropy would include another term having 
the product grad/>*grad T in the integrand. Since the latter might be either 
positive or negative, the time derivative of the entropy would not necessarily 
be positive, which is impossible. 

Finally, the above arguments must also be refined in the following respect. 
Strictly speaking, in a system which is not in thermodynamic equilibrium, 
such as a fluid with velocity and temperature gradients, the usual definitions 
of thermodynamic quantities are no longer meaningful, and must be modified. 
The necessary definitions are, firstly, that p, e and v are defined as before: 
p and pe are the mass and internal energy per unit volume, and v is the 
momentum of unit mass of fluid. The remaining thermodynamic quantities 
are then defined as being the same functions of p and e as they are in 
thermal equilibrium. The entropy s = s(p, e), however, is no longer the true 
thermodynamic entropy: the integral 



/' 



sdV 



will not, strictly speaking, be a quantity that must increase with time. Never- 
theless, it is easy to see that, for small velocity and temperature gradients, $ 
is the same as the true entropy in the approximation here used. For, if there 
are gradients present, they in general lead to additional terms (besides 
s(p, e)) in the entropy. The results given above, however, can be altered only 
by terms linear in the gradients (for instance, a term proportional to the scalar 
div v). Such terms would necessarily take both positive and negative values. 
But they ought to be negative definite, since the equilibrium value $ = s(p, e) 
is the maximum possible value. Hence the expansion of the entropy in powers 
of the small gradients can contain (apart from the zero-order term) only 
terms of the second and higher orders. 

Similar remarks should have been made in §15 (cf. the first footnote to that 
section), since the presence of even a velocity gradient implies the absence of 
thermodynamic equilibrium. The pressure p which appears in the expression 
for the momentum flux density tensor in a viscous fluid must be taken to be 
the same function p = p(p, e) as in thermal equilibrium. In this case p 
will not, strictly speaking, be the pressure in the usual sense, viz. the normal 
force on a surface element. Unlike what happens for the entropy (see 
above), there is here a resulting difference of the first order with respect to 
the small gradient; we have seen that the normal component of the force 
includes, besides p, a term proportional to div v (in an incompressible fluid, 
this term is zero, and the difference is then of higher order). 

Thus the three coefficients 17, £, k which appear in the equations of 
motion of a viscous conducting fluid completely determine the fluid-mechani- 
cal properties of the fluid in the approximation considered (i.e. when the 
higher-order space derivatives of velocity, temperature, etc. are neglected). 
The introduction of any further terms (for example, the inclusion in the mass 
flux density of terms proportional to the gradient of density or temperature) 



188 Thermal Conduction in Fluids §50 

has no physical meaning, and would mean at least a change in the definition 
of the basic quantities; in particular, the velocity would no longer be the 
momentum of unit mass of fluid. f 

§50. Thermal conduction in an incompressible fluid 

The general equation of thermal conduction (49.4) or (49.5) can be con- 
siderably simplified in certain cases. If the fluid velocity is small compared 
with the velocity of sound, the pressure variations occurring as a result of 
the motion are so small that the variation in the density (and in the other 
thermodynamic quantities) caused by them may be neglected. However, a 
non-uniformly heated fluid is still not completely incompressible in the 
sense used previously. The reason is that the density varies with the tem- 
perature; this variation cannot in general be neglected, and therefore, even 
at small velocities, the density of a non-uniformly heated fluid cannot be 
supposed constant. In determining the derivatives of thermodynamic 
quantities in this case, it is therefore necessary to suppose the pressure con- 
stant, and not the density. Thus we have 

8s I 8s \ 8T I 8s \ 

It = \8f) v ~8~t y gra Sz= l"ar/ p gra ' 

and, since T(ds/8T) p is the specific heat at constant pressure c py we obtain 
T8sj8t = c p 8T/8t, Tgrads = c p grad T. Equation (49.4) becomes 

/ 8T \ 8vi 
pc p \ + vgrad T = div(« grad T) + a' ik . (50.1) 

\ 8t / 8xjc 

If the density is to be supposed constant in the equations of motion for 
a non-uniformly heated fluid, it is necessary that the fluid velocity should be 
small compared with that of sound, and also that the temperature differences 
in the fluid should be small. We emphasise that we mean the actual values of 
the temperature differences, not the temperature gradient. The fluid may 
then be supposed incompressible in the usual sense ; in particular, the equation 
of continuity is simply div v = 0. Supposing the temperature differences 
small, we neglect also the temperature variation of r), k and c p , supposing 
them constant. Writing the term o'yc dvt/dxjc as in (49.5), we obtain the 



t Worse still, the inclusion of such terms may violate the necessary conservation laws. It must 
be borne in mind that, whatever the definitions used, the mass flux density j must always be the 
momentum of unit volume of fluid. For j is denned by the equation of continuity, 

d P /8t + div} = 0; 

multiplying this by r and integrating over the fluid volume, we have 



d(j P rdV)/dt = jjdV, 



and since the integral Jpr dV determines the position of the centre of mass, it is clear that the integral 
J" j dV is the momentum. 



§50 Thermal conduction in an incompressible fluid 189 

equation of heat transfer in an incompressible fluid in the following com- 
paratively simple form: 

8T v I dvi dvir\ 2 

— + v.gradr = x Ar+— — + — ), (50.2) 

where v — f\\p is the kinematic viscosity, and we have written k in terms of 
the thermometric conductivity, defined as 

x = Klpc p . (50.3) 

The equation of heat transfer is particularly simple for an incompressible 

fluid at rest, in which the transfer of energy takes place entirely by thermal 

conduction. Omitting the terms in (50.2) which involve the velocity, we have 

simply 

8T/dt = x^ T - (50.4) 

This equation is called in mathematical physics the equation of thermal 
conduction or Fourier's equation. It can, of course, be obtained much more 
simply without using the general equation of heat transfer in a moving 
fluid. According to the law of conservation of energy, the amount of heat 
absorbed in some volume in unit time must equal the total heat flux into this 
volume through the surface surrounding it. As we know, such a law of 
conservation can be expressed as an "equation of continuity" for the amount of 
heat. This equation is obtained by equating the amount of heat absorbed in 
unit volume in unit time to minus the divergence of the heat flux density. 
The former is pc p dTjdt; we must take the specific heat c p , since the pressure 
is of course constant throughout a fluid at rest. Equating this to — div q 
= kAT, we have equation (50.4). 

It must be mentioned that the applicability of the thermal conduction 
equation (50.4) to fluids is actually very limited. The reason is that, in 
fluids in a gravitational field, even a small temperature gradient usually 
results in considerable motion (convection; see §56). Hence we can actually 
have a fluid at rest with a non-uniform temperature distribution only if the 
direction of the temperature gradient is opposite to that of the gravitational 
force, or if the fluid is very viscous. Nevertheless, a study of the equation 
of thermal conduction in the form (50.4) is very important, since processes of 
thermal conduction in solids are described by an equation of the same form. 
We shall therefore consider it in more detail in §§51 and 52. 

If the temperature distribution in a non-uniformly heated medium at rest 
is maintained constant in time (by means of some external source of heat), 
the equation of thermal conduction becomes 

AT=0. (50.5) 

Thus a steady temperature distribution in a medium at rest satisfies Laplace's 
equation. In the more general case where k cannot be regarded a constant, 
we have in place of (50.5) the equation 

div(/c grad T) = 0. (50.6) 



190 Thermal Conduction in Fluids §50 

If the fluid contains external sources of heat (for example, heating by 
an electric current), the equation of thermal conduction must correspondingly 
contain another term. Let Q be the quantity of heat generated by these 
sources in unit volume of the fluid per unit time; Q is, in general, a function 
of the co-ordinates and of the time. Then the heat balance equation, i.e. 
the equation of thermal conduction, is 

P c p dT/dt= kAT+Q. (50.7) 

Let us write down the boundary conditions on the equation of thermal 
conduction which hold at the boundary between two media. First of all, the 
temperatures of the two media must be equal at the boundary: 

7i = T 2 . (50.8) 

Furthermore, the heat flux out of one medium must equal the heat flux 
into the other medium. Taking a co-ordinate system in which the part of the 
boundary considered is at rest, we can write this condition as/ci grad 2V df 
= k 2 grad T 2 • df for each surface element df. Putting grad T» df = (dTjdrfdf, 
where BTjdn is the derivative of T along the normal to the surface, we obtain 
the boundary condition in the form 

/ci 07i/a» = K 2 8T 2 l8n. (50.9) 

If there are on the surface of separation external sources of heat which 
generate an amount of heat £) (s) on unit area in unit time, then (50.9) must 
be replaced by 

K 1 dT 1 l8n-K 2 dT 2 ldn = £)<«>. (50.10) 

In physical problems concerning the distribution of temperature in the 
presence of heat sources, the strength of the latter is usually given as a 
function of temperature. If the function Q{T) increases sufficiently rapidly 
with T, it may be impossible to establish a steady temperature distribution 
in a body whose boundaries are maintained in fixed conditions (e.g. at a given 
temperature). The loss of heat through the outer surface of the body is 
proportional to some mean value of the temperature difference T— To between 
the body and the external medium, regardless of the law of heat generation 
within the body; it is clear that, if the generation of heat increases sufficiently 
rapidly with temperature, the loss of heat may be inadequate to achieve an 
equilibrium state. 

The impossibility of establishing a steady thermal state forms the basis of 
the thermal theory of explosions developed by N. N. Semenov (1928): if the 
rate of an exothermic combustion reaction increases sufficiently rapidly with 
temperature, the impossibility of a steady distribution leads to a rapid 
non-steady ignition of the substance and an acceleration of the reaction into a 
thermal explosion. A quantitative theory, for the case where the heat 



§50 Thermal conduction in an incompressible fluid 191 

generation is an exponential function of temperature, has been given by 
D. A. Frank-Kamenetski! (see Problem l).f 

PROBLEMS 

Problem 1. Heat sources of strength Q = Q tt e cciT - T J per unit volume are distributed in a 
layer of material bounded by two parallel infinite planes, which are kept at a constant tem- 
perature T . Find the condition for a steady temperature distribution to be possible. 

Solution. The equation for steady heat conduction is here 

K&Tldx* = -£ oC «<ZMr.), 

with the boundary conditions T = T for x = and x = 21 (21 being the thickness of the 
layer). We introduce the dimensionless variables r = ct(T—T ) and £ = xjl. Then 

t" + A<* = 0, A = £ aZ 2 //o 

Integrating this equation once (after multiplying by 2t'), we find 

t'2 = 2\{e^-e% 

where t is a constant, which is evidently the maximum value of t; by symmetry, this value 
must be attained half-way through the layer, i.e. for £ — 1. Hence a second integration, with 
the condition t = for £ = 0, gives 

1 r &r r 

V(2A) J V^-er) = J df==1 ' 

Effecting the integration, we have 

e~* T . cosh-M r . = \Z(i*). (1) 

The function A(t ) determined by this equation has a maximum A = A cr for a definite value 
T o — T o.or>" if A > A CT , there is no solution satisfying the boundary conditions.^ The numerical 
values are A„ = 0-88, T 0>cr = 1 '2.ft 

Problem 2. A sphere is immersed in a fluid at rest, in which a constant temperature 
gradient is maintained. Determine the resulting steady temperature distribution in the fluid 
and the sphere. 

Solution. The temperature distribution satisfies the equation A T = in all space, with 
the boundary conditions 

7i = T 2 , /ci dTxjdr = k 2 dT 2 /dr 

for r = R (where R is the radius of the sphere ; quantities with the suffixes 1 and 2 refer to 
the sphere and the fluid respectively), and grad T = A at infinity, where A is the given 

f The rate of explosive combustion reactions, and therefore the rate of heat generation, depend 
on temperature roughly as e - v l BT , the constant U being large. Frank- KamenetskiI has shown that, 
to investigate the conditions for a thermal explosion to occur, we must consider the course of the 
reaction when the ignition of the substance is comparatively slow, and therefore replace e~ u l BT by 
e-uvir, e v(T-T tt )iRT a * y w here T is the external temperature. A more detailed discussion is given in the 
book by D. A. Frank- KamenetskiI, Diffusion and Heat Exchange in Chemical Kinetics Princeton 
1955. 

% Only the smaller of the two roots of equation (1) for A < Acr corresponds to a stable temperature 
distribution. 

ff The corresponding values for a spherical region (of radius Z) are Acr — 3-32, T 0>C r — 1-47, and for 
an infinite cylinder A cr = 2- 00, T 0lC r = 1-36. 



192 Thermal Conduction in Fluids §51 

temperature gradient. By the symmetry of the problem, A is the only vector which can 
determine the required solution. Such solutions of Laplace's equation are constant X At 
and constant X A • grad(l /r). Noticing also that the solution must remain finite at the centre 
of the sphere, we seek the temperatures T x and T 2 in the forms 

7i = Cl A.r, T 2 = c 2 A.r/r3+A.r. 

The constants c t and c 2 are determined from the conditions for r = R, the result being 
3/C2 I" K2 — K1 / R \ 3 1 



Ti = 



'- »-['♦££(')>« 



K± + 2/C2 

§51. Thermal conduction in an infinite medium 

Let us consider thermal conduction in an infinite medium at rest. The 
most general problem of this kind is as follows. The temperature distribution 
is given in all space at the initial instant t = : 

T = To(x,y,z) for t = 0, 

where To is a given function of the co-ordinates. It is required to determine 
the temperature distribution at all subsequent instants. 

We expand the required function T as a Fourier integral with respect to 
the co-ordinates: 

T = j 7k(*)exp(*-k.r)d3k, d3k = <\k x dk y dk z , (51.1) 

where the expansion coefficients are given by 

n{t) = (2tt)-3 J r(^/,s',*) ex P(-^"-rW', dV = dx'&y'&z'. 

Substituting the expression (51.1) in equation (50.4), we obtain 
dTk 



f l--^- + & 2 xrjexp(*-k.r)d3k = 0, 



whence 

dTJdt+kz x T k = 0. 
This equation gives T k as a function of time : 

T k = exp(— k 2 xt)T 0k . 
Substituting this in (51.1), we find 

T = J" r 0k exp(-#y) exp(/k.r)d3k. (51.2) 

Since we must have T = To(x, y, z) for / = 0, it is clear that the 
Tok are the expansion coefficients of the function To(x, y, z) as a Fourier 
integral: 

T 0k = (2tt)-3 J r (^/,*')exp(-tk.r')dF. 



§51 Thermal conduction in an infinite medium 193 

Finally, substituting this in (51.2), we obtain 

T = (2tt)-3 J J T (x',y',z') exp(-^) exp\ik.(r-r')]dV &k. 
The integral over k is the product of three simple integrals, each of the form 

00 00 

J exp( — k x 2 xt) exp [ik x (x — x')] dk x = J exp( — k x 2 xt) cos k x (x —x') dk x ; 

—00 —00 

the similar integral with sin in place of cos is zero, since the sine function 
is odd. Using the formula 

00 

I exp( — ax 2 ) cos /tod*; = ■\/( 7T / a ) ex P( — /? 2 /4a) (a > 0), 

—oo 

we have finally 

T(x,y,z,t) = — — - JV (*', /, *') x 

exp{- [{x - x'f + (y -y'f + (z- z'f]l4 x t} dV. (51.3) 

This formula gives the complete solution of the problem ; it determines the 
temperature distribution at any instant in terms of the given initial distri- 
bution. 

If the initial temperature distribution is a function of only one co-ordinate, 
x, then we can integrate over y' and z' in (51.3) and obtain 

T(x, t) = —-— T 7o(*') exp[-(*-*') 2 /4x'] d*'. (51.4) 

VW) -oo 

At time t = 0, let the temperature be zero in all space except for an infinitely 
thin layer at the plane x = 0, where it is infinite in such a way that the total 
quantity of heat (proportional to $To(x)dx) is finite. Such a distribution can 
be represented by a delta function: To(x) = constant x8(x). The integration 
in formula (51.4) then amounts to replacing x' by zero, the result of which is 

1 
T(x, t) = constant x exp( - # 2 /4x*)- (51.5) 

Similarly, if at the initial instant a finite quantity of heat is concentrated 
at a point (the origin), the temperature distribution at subsequent instants 
is given by the formula 

1 
T(r, t) = constant x — — -exp( - r 2 /4 x t), (51.6) 

S{7Txty 

where r is the distance from the origin. In the course of time, the temperature 



194 



Thermal Conduction in Fluids 



§51 



at the point r = decreases as H. The temperature in the surrounding 
space rises correspondingly, and the region where the temperature is appre- 
ciably different from zero expands (Fig. 29). The manner of this expansion 
is determined principally by the exponential factor in (51.6). We see that 
the order of magnitude / of the dimension of this region is given by I 2 fat ~ 1, 
whence 



i ~ V(xt), 

i.e. / increases as the square root of the time. 



(51.7) 




Formula (51.7) can also be interpreted in a somewhat different way. Let / 
be the order of magnitude of the dimension of a body. Then we can say that, 
if the body is heated non-uniformly, the order of magnitude r of the time 
required for the temperature to become more or less the same throughout 
the body is 

r ~ J 2 /*. (51.8) 

The time t, which may be called the relaxation time for thermal conduction, 
is proportional to the square of the dimension of the body, and inversely 
proportional to the thermometric conductivity. 

The thermal conduction process described by the formulae obtained above 
has the property that the effect of any perturbation is propagated instan- 
taneously through all space. It is seen from formula (51.5) that the heat 



§51 Thermal conduction in an infinite medium 195 

from a point source is propagated in such a manner that, even at the next 
instant, the temperature of the medium is zero only at infinity. This property 
holds also for a medium in which the thermometric conductivity x depends on 
the temperature, provided that x does not vanish anywhere. If, however, x 
is a function of temperature which vanishes when T = 0, the propagation of 
heat is retarded, and at each instant the effect of a given perturbation extends 
only to a finite region of space (we suppose that the temperature outside this 
region can be taken as zero). This result, as well as the solution of the 
following Problems, is due to Ya. B. Zel'dovich and A. S. Kompaneets 
(1950). 

PROBLEMS 

Problem 1. The specific heat and thermal conductivity of a medium vary as powers of the 
temperature, while its density is constant. Determine the manner in which the tempera- 
ture tends to zero near the boundary of the region which, at a given instant, has received 
heat propagated from an arbitrary source (the temperature outside that region being zero). 

Solution. If k and c P vary as powers of the temperature, the same is true of the thermo- 
metric conductivity x and of the heat function 



ZO 



= j CpdT 



(we omit a constant in w). Hence we can put x — aW n , where we denote by W = pw the 
heat function per unit volume. Then the thermal conduction equation 

pc p BTjdt = div(*c grad T) 

becomes 

dW/dt = a di\(Wn grad W). (1) 

During a short interval of time, a small portion of the boundary of the region may be 
regarded as plane, and its rate of displacement in space, v, may be supposed constant. Accord- 
ingly, we seek a solution of equation (1) in the form W = W(x —vt), where x is the co-ordinate 
in the direction perpendicular to the boundary. We have 

-vdWjdx = ad(W« dW/dx)ldx, (2) 

whence we find, after two integrations, that W vanishes as 

W ~ |*|i/» (3) 

where \x\ is the distance from the boundary of the heated region. This also confirms our 
conclusion that, if n > 0, the heated region has a boundary outside which W and T are 
zero. If n < 0, then equation (2) has no solution vanishing at a finite distance, i.e. the heat 
is distributed through all space at every instant. 

Problem 2. A medium like that described in Problem 1 has, at the initial instant, an amount 
of heat Q per unit area concentrated on the plane * = 0, while T = everywhere else. Deter- 
mine the temperature distribution at subsequent instants. 

Solution. In the one-dimensional case, equation (1) of Problem 1 is 

dw a / 8W\ 

= a —\Wn . (1) 

dt dx\ 8x ] 

From the parameters Q and a and variables * and t at our disposal, we can form only one 
dimensionless combination, 

£ = x/(Qnat)W+M; (2) 



196 Thermal Conduction in Fluids §52 

Q and a have the dimensions erg/cm 8 and (cm a /sec)(cm 3 /erg)». Hence the required function 
W(x, t) must be of the form 

^=(2 2 M 1/(2+ »m (3) 

where the dimensionless function /(0 is multiplied by a quantity having the dimensions 
erg/cm 3 . With this substitution, equation (1) gives 



d / d/\ dY 



This ordinary differential equation has a simple solution which satisfies the conditions of the 
problem, namely 

f(i) = [M& 2 -| 2 )/(2 + n)]i/», ( 4) 

where £ is a constant of integration. 

For n > 0, this formula gives the temperature distribution in the region between the planes 
x = ±* corresponding to the equation £ = ±$ ; outside this region, W = 0. Hence it 
follows that the heated region expands with time in a manner given by x = constant X i x /( 2 +»). 
The constant £ is determined by the condition that the total amount of heat is constant: 

Xo £o 

Q- j Wdx = Q J7(fld£ (5) 

— #0 ~~ So 



whence we have 



= (2+^21-* r»(j+i/*») 

n7T n/2 r»(l/w) ' w 

For n — — v < 0, we write the solution in the form 

Here the heat is distributed through all space, and at large distances W decreases as x~ ilv . 
This solution is valid only for v < 2 ; for v > 2, the normalisation integral (5) (which now 
extends to ± oo) diverges, which means physically that the heat is conducted instantaneously 
to infinity. For v < 2, the constant £ in (7) is given by 

2(2-„k/2 r-(i/ y -j) 

h = — ; — j^m- W 

Finally, f or n -> we have £ -»- 2/V«, and the solution given by formula (3) of Problem 1 
(1), and (4) is ' 

( Q l x 2 \ 1/n \ o 

w - feferw 1 = 2^br*-« 

in agreement with formula (51.5). 

§52. Thermal conduction in a finite medium 

In problems of thermal conduction in a finite medium, the initial tempera- 
ture distribution does not suffice to determine a unique solution, and the 
boundary conditions at the surface of the medium must also be given. 



§52 Thermal conduction in a finite medium 197 

Let us consider thermal conduction in a half-space (x > 0), beginning 
with the case where a given constant temperature is maintained on the 
bounding plane x = 0. We may arbitrarily take this temperature as zero. 
At the initial instant, the temperature distribution throughout the medium is 
given, as before. The boundary and initial conditions are therefore 

T = for * = 0; T = T (x,y,z) for t = and * > 0. (52.1) 

The solution of the thermal conduction equation with these conditions can, 
by means of the following device, be reduced to the solution for a medium 
infinite in all directions. We imagine the medium to extend on both sides of 
the plane x = 0, the temperature distribution for t = and x < being 
given by — To. That is, the temperature distribution at the initial instant 
is given in all space by an odd function of x : 

?o( - x, y, z)= - T (x,y, z). (52.2) 

It follows from equation (52.2) that 7o(0, y, z) = — 7o(0, y, z) = 0, i.e. the 
necessary boundary condition (52.1) is automatically satisfied for t = 0, 
and it is evident from symmetry that it will continue to be satisfied for all t. 
Thus the problem is reduced to the solution of equation (50.4) in an 
infinite medium with an initial function To(x,y, z) which satisfies (52.2), 
and without boundary conditions. Hence we can use the general formula 
(51.3),. We divide the range of integration over x' in (51.3) into two parts, 
from — oo to and from to oo. Using the relation (52.2), we then have 



OO OO 00 



r(w<) = iiiJJI r ° (W): 



—oo—oo 

{exp[-(x—x'f/4xt]-exp[-(x+x') 2 l4xt]} x 

exp { - [(y - y'f + (z- z'f]l4 x t] oV Ay' dz'. (52.3) 

This formula gives the solution of the problem, since it determines the tem- 
perature throughout the medium, i.e. for all x > 0. 

If the initial temperature distribution is a function of x only, formula 
(52.3) becomes 

1 °° 
T (*'') = ^TTT^ \ 71 o(^){exp[-(^-^)2 / 4^ ] _e X p[_^ + ^)2/4^]}d^. 

(52.4) 

As an example, let us consider the case where the initial temperature is a 
given constant everywhere except at x = 0. Without loss of generality, this 
constant may be taken as — 1. The temperature on the plane x = is always 
zero. The appropriate solution is obtained at once by substituting 



198 



Thermal Conduction in Fluids 



§52 



Tq{x) = — 1 in (52.4). The integral in (52.4) is the sum of two integrals, in 
each of which we change the variables as in £ = (x' — x)/2^(xt). We then 
obtain for T(x, t) the expression 

T{x,t) = Kerf[-*/2V(*0]-erf[*/2V(xO]}> 
where the function erf x is defined as 



2 r , 

erf# = £~^d£, 

\f-n J 



(52.5) 



and is called the error function (we notice that erf oo = 1). Since erf ( — x) 
— — erf x, we have finally 

T{x y t) = - erf [x/2V( x t)]. (52.6) 

Fig. 30 shows a graph of the function erf x. The temperature distribution 
becomes more uniform in space in the course of time. This occurs in such a 



VO 
0-8 














— 




























l. 
<u 

0-4 
0-2 































































O 0-2 0-4 0-6 0-8 1-0 1-2 1-4 1-6 1-8 2"0 

x 
Fig. 30 



way that any given value of the temperature "moves" proportionally to \/t. 
This last result is obviously true. For the problem under consideration is 
characterised by only one parameter, the initial temperature difference Tq 
between the boundary plane and the remaining space; in the above discussion, 
this difference was arbitrarily taken as unity. From the parameters To 
and x and variables x and t at our disposal we can form only one dimension- 
less combination, xj^/{xf)'y hence it is clear that the required temperature 
distribution must be given by a function of the form T = To/(a:/\/(xO)' 

Let us now consider a case where the surface bounding the medium is a 
thermal insulator. That is, there is no heat flux at the plane x = 0, so that 
we must have dTjdx = 0. We thus have the following boundary and initial 



§52 Thermal conduction in a finite medium 199 

conditions : 

dT/dx = for x = 0; T = T (x,y,z) for t = 0, * > 0. (52.7) 

To find the solution we proceed as in the previous problem. That is, we again 
imagine the medium to extend on both sides of the plane x = 0, the initial 
temperature distribution being this time symmetrical about the plane. In 
other words, we now suppose that To(x, y, z) is an even function of x: 

T ( - x,y, z) = T (x,y, z). (52.8) 

Then dT (x,y, z)\Bx = -dT (-x,y, z)/dx, and dT Q jdx = for x = 0. It 
is evident from symmetry that this condition will continue to be satisfied 
for all t. 

Repeating the calculations given above, but using (52.8) in place of (52.2), 
we have the general solution of the problem in the form 

00 00 00 



r(w) = i^/J7 r °^> 



—oo—oo 

{exp[- (x' - xf/4 x t] + exp[- (*' + xfjAxt]} x 
exp{ - [(/ -yf + (z' - zf]l4 x t} dx' dy'dz'. (52.9) 

If To is a function of x only, then 

T(*>t) = 2^—^ J ^'){exp[-(^-^/4^] + exp[-(^ + ^/4^]}d^'. 
(52.10) 

Let us now consider problems with boundary conditions of a different type, 
which also enable the equation of thermal conduction to be solved in a general 
form. Let a heat flux (a given function of time) enter a medium through its 
bounding plane x = 0. The boundary and initial conditions are 

-k8T/8x = q(t) for * = 0; T == for t = - oo, x > 0, (52.11) 

where q(t) is a given function. 

We first solve an auxiliary problem, in which q(t) = 8(t). It is easy to 
see that this problem is physically equivalent to that of the propagation of 
heat in an infinite medium from a point source which generates a given 
amount of heat. For the boundary condition - kBTjBx = 8(t) for x = 
signifies physically that a unit of heat enters through each unit area of the 
plane x = at the instant t = 0. In the problem where the condition is 
T — 28(x)lpc P for t = 0, an amount of heat 

J pc p Tdx = 2 

is concentrated on this area at time t = 0; half of this is then propagated in 



200 Thermal Conduction in Fluids §52 

the positive ^-direction, and the other half in the negative ^-direction. 
Hence it is clear that the solutions of the two problems are identical, and we 
find from (51.5) kT(x, t) = V(xH) exp( -*a/4tf)- 

Since the equations are linear, the effects of the heat entering at different 
moments are simply additive, and therefore the required general solution of 
the equation of thermal conduction with the conditions (52.11) is 

t 
kT(x, t)= j y_A_ ? ( T ) exp[-*2/4 x (*_ T )] dr. (52.12) 

—oo 

In particular, the temperature on the plane x = varies according to 

t 

Kr(o -< )= jy^ (T)dT - (52 - i3) 

—oo 

Using these results, we can obtain at once the solution of another problem, 
in which the temperature T on the plane x = is a given function of time: 

T = T Q (t) for x = 0; T = for t = - oo, x > 0. (52.14) 

To do so, we notice that, if some function T(x, t) satisfies the equation of 
thermal conduction, then so does its derivative dT/dx. Differentiating 
(52.12) with respect to x, we obtain 

t 
8T(x, t) p x i{ r ) 

— K 



= l 2VrXrW a '*-*W-r)]dr. 



8x J 2VI>x(*-t) 3 ] 

—oo 

This function satisfies the equation of thermal conduction and (by (52.11)) 
its value for x = is q(t); it therefore gives the required solution of the 
problem whose conditions are (52.14). Writing T(x, t) instead of — icdT/dx, 
and To(t) instead of q(t), we thus have 

t 
T ^ f > = TTT^ \ TT^Pt-* 8 /^'-^ ^ (52.15) 

—oo 

The heat flux q = — kBT/Bx through the bounding plane x = is found by 
a simple calculation to be 



t 



k r dr (T) dr 



—00 

This formula is the inverse of (52.13). 



§52 Thermal conduction in a finite medium 201 

The solution is easily obtained for the important problem where the 
temperature on the bounding plane x = is a given periodic function of 
time: T = Toe~ ib)t for x = 0. It is clear that the temperature distribution 
in all space will also depend on the time through a factor e~ M . Since the 
one-dimensional equation of thermal conduction is formally identical with 
the equation (24.3) which determines the motion of a viscous fluid above an 
oscillating plane, we can immediately write down the required temperature 
distribution by analogy with (24.4) : 

T = T exp[-*V(W 2 x)] exp{i[xV{(o(2 x )-cot]}. (52.17) 

We see that the oscillations of the temperature on the bounding surface are 
propagated from it as thermal waves which are rapidly damped in the interior 
of the medium. 

Another kind of thermal-conduction problem comprises those concerning 
the rate at which the temperature is equalised in a non-uniformly heated 
finite body whose surface is maintained in given conditions. To solve these 
problems by general methods, we seek a solution of the equation of thermal 
conduction in the form T = T n (x, y, z)e~^nt y w ith X n a constant. For the 
function T n we have the equation 

xAT n = -A n 7V (52.18) 

This equation, with given boundary conditions, has non-zero solutions only 
for certain \ n , its eigenvalues. All the eigenvalues are real and positive, 
and the corresponding functions T n (x,y, z) form a complete set of orthogonal 
functions. Let the temperature distribution at the initial instant be given by 
the function To(x, y, z). Expanding this as a series of functions T n , 

T (x,y,z) = ^c n T n (x,y,z), 

we obtain the required solution in the form 

T(x, y, z, t) = ^c n T n (x,y, z) exp(-X n t). (52.19) 

The rate of equalisation of the temperature is evidently determined by the 
term corresponding to the smallest A n , which we call Ai. The "equalisation 
time" may be defined as t = 1/Ai. 

PROBLEMS 

Problem 1. Determine the temperature distribution around a spherical surface (of radius 
R) whose temperature is a given function T (t) of time. 

Solution. The thermal-conduction equation for a centrally symmetrical temperature distri- 
bution is, in spherical co-ordinates, dTJdt = (x/r)3 8 (rr)/^r 2 . The substitution rT(r, t) 
= F(r, t) reduces this to dF/dt = xd^FIdr*, which is the ordinary one-dimensional thermal- 
conduction equation. Hence the required solution can be found at once from (52.15), and is 

R(r~R) r T (r) 

2r VM J (t-ry 



202 Thermal Conduction in Fluids §53 

Problem 2. The same as Problem 1, but for the case where the temperature of the spherical 
surface is T e~ <wt . 

Solution. Similarly to (52.17), we obtain 

T = T exp(-icot)(Rlr) exp[-(l-i)(r-RW(co/2x)]. 

Problem 3. Determine the temperature equalisation time for a cube of side a whose 
surface is (a) maintained at a temperature T = 0, (b) an insulator. 

Solution. In case (a) the smallest value of A is given by the following solution of equation 
(52.18): 

T\ = sm(7Tx/a) s'mfry/a) s\n{iTzJa) 

(the origin being at one corner of the cube), when r = 1/A X «= a z J3w*x- In case (b) we have 
Z 1 ! = cos(7rxJa) (or the same function of y or z), when r = a 2 Jn 2 x- 

Problem 4. The same as Problem 3, but for a sphere of radius R. 

Solution. The smallest value of A is given by the centrally symmetrical solution of (52.18) 
Ti = (1/r) sin kr; in case (a), k — nJR, and r = l/ x k* = R 2 Jx**- In case (b) k is the 
smallest non-zero root of the equation kR = tan kR, whence we find kR = 4-493 and 
t = 0-050 R*/ X - 

§53. The similarity law for heat transfer 

The processes of heat transfer in a fluid are more complex than those in 
solids, because the fluid may be in motion. A heated body immersed in a 
moving fluid cools considerably more rapidly than one in a fluid at rest, where 
the heat transfer is accomplished only by conduction. The motion of a 
non-uniformly heated fluid is called convection. 

We shall suppose that the temperature differences in the fluid are so small 
that its physical properties may be supposed independent of temperature, 
but are at the same time so large that we can neglect in comparison with them 
the temperature changes caused by the heat from the energy dissipation 
by internal friction (see §55). Then the viscosity term in equation (50.2) may 
be omitted, leaving 

ar/S/+ vgrad T = X AT, (53.1) 

where x = KJpc P is the thermometric conductivity. This equation, together 
with the Navier-Stokes equation and the equation of continuity, completely 
determines the convection in the conditions considered. 

In what follows we shall be interested only in steady convective flow.f 
Then all the time derivatives are zero, and we have the following fundamental 
equations : 

v.gradr = x AT, (53.2) 

(v»grad)v = -grad(p/p) + vAv, divv = 0. (53.3) 



t In order that the convection should be steady, it is, strictly speaking, necessary that the solid 
bodies adjoining the fluid should contain sources of heat which maintain these bodies at a constant 
temperature. 



§53 The similarity law for heat transfer 203 

This system of equations, in which the unknown functions are v, T and pfp, 
contains only two constant parameters, v and x- Furthermore, the solution 
of these equations depends also, through the boundary conditions, on some 
characteristic length /, velocity U, and temperature difference T\— To. 
The first two of these are given by the dimension of the solid bodies which 
appear in the problem and the velocity of the main stream, while the third 
is given by the temperature difference between the fluid and these bodies. 

In forming dimensionless quantities from the parameters at our disposal, 
the question arises of the dimensions to be ascribed to the temperature. To 
resolve this, we notice that the temperature is determined by equation (53.2), 
which is linear and homogeneous in T. Hence the temperature can be multi- 
plied by any constant and still satisfy the equations. In other words, the unit 
of measurement of temperature can be chosen arbitrarily. The possibility 
of this transformation of the temperature can be formally allowed for by 
giving it a dimension of its own, unrelated to those of the other quantities. 
This can be measured in degrees, the usual unit of temperature. 

Thus convection in the above-mentioned conditions is characterised by 
five parameters, whose dimensions are v = x = cm 2 /sec, U = cm/sec, 
/ = cm, 7i— To = deg. From these we can form two independent dimen- 
sionless combinations. These may be the Reynolds number R = Uljv 
and the Prandtl number, defined as 

P = vlx. (53.4) 

Any other dimensionless combination can be expressed in terms of R and P.f 
The Prandtl number is just a constant of the material, and does not depend 
on the properties of the flow. For gases it is always of the order of unity. 
The value of P for liquids varies more widely. For very viscous liquids, it 
may be very large. The following are values of P at 20°C for various 
substances : 

Air 0-733 

Water 6-75 

Alcohol 16-6 

Glycerine 7250 

Mercury 0-044 

As in §19, we can now conclude that, in steady convection (of the type 
described), the temperature and velocity distributions are of the form 

T-Tq It \ v It \ 

lwr / (? R 4 u = t (-A (53 - 5) 

The dimensionless function which gives the temperature distribution depends 
on both R and P as parameters, but the velocity distribution depends only on 
R, since it is determined by equations (53.3), which do not involve the con- 
ductivity. Two convective flows are similar if their Reynolds and Prandtl 
numbers are the same. 



t The Peclet number is sometimes used; it is defined as UlJx = RP. 



204 Thermal Conduction in Fluids §53 

The heat transfer between solid bodies and the fluid is usually characterised 
by the heat transfer coefficient a, defined by 

a = ^/(Ti-r ), (53.6) 

where q is the heat flux density through the surface and T\- To is a charac- 
teristic temperature difference between the solid body and the fluid. If 
the temperature distribution in the fluid is known, the heat transfer coefficient 
is easily found by calculating the heat flux density q = - K 8Tjdn at the 
boundary of the fluid (the derivative being taken along the normal to the 
surface). 

The heat transfer coefficient is not a dimensionless quantity. A dimension- 
less quantity which characterises the heat transfer is what is called the Nusselt 
number :f 

N = oJ/k. (53.7) 

It follows from similarity arguments that, for any given type of convective 
flow, the Nusselt number is a definite function of the Reynolds and Prandtl 
numbers only: 

N=/(R,P). (53.8) 

This function is very simple for convection at sufficiently small Reynolds 
numbers. These correspond to small velocities. Hence, in the first approxi- 
mation, we can neglect the velocity term in equation (53.2), so that the 
temperature distribution is determined by the equation AT = 0, i.e. the 
ordinary equation of steady thermal conduction in a medium at rest. The heat 
transfer coefficient can then depend on neither the velocity nor the viscosity 
and so we must have simply 

N = constant, (53.9) 

and in calculating the constant the fluid may be supposed at rest. 

PROBLEM 

Determine the temperature distribution in a fluid moving in Poiseuille flow along a pipe 
of circular cross-section, when the temperature of the walls varies linearly along the pipe. 

Solution. The conditions of the flow are the same at every cross-section of the pipe, and 
we can look for the temperature distribution in the form T — Az+f(r), where Az is the wall 
temperature; we use cylindrical co-ordinates, with the x-axis along the axis of the pipe. 
For the velocity we have, by (17.9), v z = v = 2v m {\ —r 2 jR 2 ), where v m is the mean velocity. 
Substituting in (53.2) we find 

1 d / d/\ 2VmA 



1 d/d/X IVmAV /,yn 
rdr\dr) x |_ \R/1' 



The solution of this equation which is finite for r = and zero for r = R is 

VmAr 2 



f(r) = 



2 X 



[-(iHGH- 



f The dimensionless "heat transfer number", denned as Kh ■» ccjpcpU = N/RP, is also used. 



§54 Heat transfer in a boundary layer 205 

The heat flux density is 

q = K [dT/dr] R = \pc v v m RA. 
It is independent of the thermal conductivity. 

§54. Heat transfer in a boundary layer 

The temperature distribution in a fluid at very high Reynolds numbers 
exhibits properties similar to those of the velocity distribution. Very large 
values of R are equivalent to a very small viscosity. But since the number 
P = v fx is not small, the thermometric conductivity x must be supposed 
small, as well as v. This corresponds to the fact that, for sufficiently high 
velocities, the fluid may be approximately regarded as an ideal fluid, and in 
an ideal fluid both internal friction and thermal conduction are absent. 

This viewpoint, however, must again be abandoned in a boundary layer, 
since neither the boundary condition of no slip nor that of equal temperatures 
would be satisfied. In the boundary layer, therefore, there occurs both a 
rapid decrease of the velocity and a rapid change of the fluid temperature to a 
value equal to the temperature of the solid surface. The boundary layer is 
characterised by the presence of large gradients of both velocity and tem- 
perature. 

It is easy to see that, in flow past a heated body (with R large), the 
heating of the fluid occurs almost exclusively in the wake, while outside the 
wake the fluid temperature does not change. For, when R is large, the pro- 
cesses of thermal conduction in the main stream are unimportant. Hence the 
temperature varies only in the region reached by fluid that has been heated 
in the boundary layer. We know (see §34) that the streamlines from 
the boundary layer enter the main stream only beyond the line of separation, 
where they go into the region of the turbulent wake. From the wake, however, 
the streamlines do not emerge at all. Thus the fluid which flows past the 
surface of the heated body in the boundary layer goes entirely into the wake 
and remains there. We see that the heat becomes distributed through the 
regions where the vorticity is non-zero. 

In the turbulent region itself, a very considerable exchange of heat occurs, 
which is due to the intensive mixing of the fluid characteristic of any turbulent 
flow. This mechanism of heat transfer may be called turbulent conduction and 
characterised by a coefficient K tuih , in the same way as we introduced the 
turbulent viscosity v tmb in §31. The turbulent thermometric conductivity 
is defined, in order of magnitude, by the same formula as v turb (31.2): 

Xturb ~ l^u. 

Thus the processes of heat transfer in laminar and in turbulent flow are 
fundamentally different. In the limiting case of very small viscosity and 
thermal conductivity, in laminar flow, the processes of heat transfer are 
absent, and the fluid temperature is constant at every point in space. In 
turbulent flow, however, even in the same limiting case, heat transfer occurs 
and rapidly equalises the temperatures in various parts of the stream. 



206 Thermal Conduction in Fluids §54 

It should be mentioned that, when we speak of the temperature of a fluid in 
turbulent motion, we mean the time average of the fluid temperature. The 
actual temperature at any point in space undergoes very irregular variations 
with time, similar to those of the velocity. 

Let us begin by considering heat transfer in a laminar boundary layer. 
The equations of motion (39.10) are unaltered. A similar simplification 
must now be performed for equation (53.2). Written explicitly, this equation 
is (since all quantities are independent of the co-ordinate z) 

dT 8T /d*T d*T\ 

Vx ~fa +Vy ~dy ~ X \lw + ~df}' 

On the right-hand side we may neglect the derivative 8 2 TJdx 2 in comparison 
with 8 2 T/dy 2 , leaving 

BT dT d*T 

vx-— + v y = x — -. (54.1) 

dx 8y A dy* v ' 

By comparing this equation with the first of (39.10) we see that, if the 
Prandtl number is of the order of unity, then the order of magnitude 8 of the 
thickness of the layer in which the velocity v x decreases and the temperature 
T varies will again be given by the formulae obtained in §39, i.e. it will be 
inversely proportional to <\/R. The heat flux q = - KdTJdn is equal, in 
order of magnitude, to k{T\ - T )/ 8. Hence we conclude that q, and therefore 
the Nusselt number, are proportional to -\/R. The dependence of N on P 
is not determined. Thus we have 

N = VR/(P). (54.2) 

From this it follows, in particular, that the heat transfer coefficient a is 
inversely proportional to the square root of the dimension / of the body. 

Let us now consider heat transfer in a turbulent boundary layer. Here it 
is convenient, as in §42, to take an infinite plane-parallel turbulent stream 
flowing along an infinite plane surface. The transverse temperature gradient 
dT/dy in such a flow can be determined from the same kind of dimensional 
argument as we used to find the velocity gradient du/dy. We denote by q 
the heat flux density along the jy-axis caused by the temperature gradient. 
This flux is a constant (independent of y), like the momentum flux a, and 
can likewise be regarded as a given parameter which determines the proper- 
ties of the flow. Furthermore, we have as parameters also the density p and 
the specific heat c p per unit mass. Instead of a we use as parameter v m \ 
q and c v have the dimensions erg/cm 2 sec = g/sec 3 and erg/g deg = cm 2 /sec 2 
deg. The viscosity and thermal conductivity cannot appear explicitly in 
dT\dy when R is sufficiently large. 

Because of the homogeneity of the equations as regards the temperature, 
already mentioned in §53, the temperature can be changed by any factor 
without violating the equations. When the temperature is changed in this 
way, however, the heat flux must change by the same factor. Hence q and T 



§54 Heat transfer in a boundary layer 207 

must be proportional. From q, v m p, c v and y we can form only one quantity 
proportional to q and having the dimensions deg/cm, namely qjpCp v m y. 
Thus we must have dT/dy = ^qjbpc v v m y, where /? is a numerical constant 
which must be determined by experiment, j* Hence 

T = (Pq/bpc p v m )(\ogy + c). (54.3) 

Thus the temperature, like the velocity, varies logarithmically. The constant 
of integration c which appears here must be determined from the conditions 
in the viscous sublayer, as in the derivation of (42.7). The temperature diff- 
erence between the fluid at a given point and the wall (which we arbitrarily 
take to be at zero temperature) is composed of the temperature change across 
the turbulent layer and that across the viscous sublayer. The logarithmic 
law (54.3) is determined by only the first of these. Hence, if we write (54.3) 
in the form T = (Pqlbpc p v m )[log(yvJv) + constant], including in the argument 
of the logarithm a factor equal to the thickness yo, then the constant (multi- 
plied by the coefficient in parentheses) must be the change in temperature 
across the viscous sublayer. This change, of course, depends on the coeffi- 
cients v and x also. Since the constant is dimensionless, it must be some 
function of P, which is the only dimensionless combination of the quantities 
v, x, p, v# and c v (q cannot appear, since T must be proportional to q, which 
already occurs in the coefficient). Thus we find the temperature distribution 
to be 

T = (pqlbpc p v*)[log(vvJv) +/(P)]. (54.4) 

Using this formula, we can calculate the heat transfer for turbulent flow in a 
pipe, along a flat plate, etc. We shall not pause to do this here. 

PROBLEMS 

Problem 1 . Determine the limiting form of the dependence of the Nusselt number on the 
Prandtl number in a laminar boundary layer when P and R are large. 

Solution. For large P, the distance 8' over which the temperature changes is small 
compared with the thickness 8 of the layer in which the velocity v x diminishes. 8' may be 
called the thickness of the temperature boundary layer. The order of magnitude of 8' may 
be obtained from an estimate of the terms in equation (54.1). Over the distance from y — 
to y f-j 8', the temperature varies by an amount of the order of the total temperature diff- 
erence T r — T between the fluid and the solid body, while the velocity v x varies over this 
distance by an amount of the order of C/S'/S (since the total change, of the order of U, occurs 
over a distance 8). Hence, for y <->' 8', the terms in equation (54.1) are, in order of magnitude, 

x d2T/dyZ ~ x (Ti- 7o)/S' 2 and v x dT/dx ~ C/S'(Ti- T )//S. 

If the two expressions are comparable, we have S' 3 "~ x/S/t7. Substituting 8 ~ l\ V-R, 
we obtain 8' ~ Z/Rip* ^/ S/Pi. Thus, for large P, the thickness of the temperature boundary 
layer decreases, relative to that of the velocity boundary layer, inversely as the cube root of P. 



f Here b is the constant appearing in the logarithmic velocity profile (42.4). With this definition, 
P is the ratio vtart>/xturb> where rturb and xturb are the coefficients in q = pc P xturbdT/dy, 
a = pvtnrbdtt/dy. From simultaneous determinations of the velocity and temperature profiles in 
pipes and in flow along flat plates,/? is found to be about 0-7. We should mention that similar measure- 
ments in the turbulent wake behind a heated body give a value of about 0- 5 for the ratio vturb/xturb in 
a free turbulent flow. 



208 Thermal Conduction in Fluids §54 

The heat flux q = — KdTJdy ~ k(T x - T )jo", and the required limiting law of heat transfer 
is found to bef 

N = constant x R*P*. 

Problem 2. Determine the limiting form of the function /(P), in the logarithmic tempera- 
ture distribution (54.4), for large values of P. 

Solution. According to what was said in §42, the transverse velocity in the viscous sub- 
layer is of the order of v*(yfy ) 2 , while the scale of the turbulence is of the order of y*fy . 
The turbulent thermometric conductivity x turb is therefore of the order of 

»*.yoCv/yo) 4 ~ v(ylyo) 4 

(where we have used the relation (42.5)); xturb is comparable in magnitude with the ordinary 
coefficient x at distances of the order of y r ~ y oP - *. Since x turb increases very rapidly with y, 
it is clear that most of the temperature change in the viscous sublayer occurs over distances 
from the wall of the order of y u and may be supposed proportional to y lf being in order of 
magnitude qyj k ~ qy f kP* ~ qP*/pc P v*. Comparing with formula (54.4), we see that the 
function /(P) is a numerical constant times P*.J 

Problem 3. Determine the temperature differences T^ in a non-uniformly heated turbulent 
fluid over distances A which are small compared with the external scale of the turbulence 
(A. M. Obukhov 1949). 

Solution. The equalisation of temperature in a non-uniformly heated turbulent fluid 
occurs similarly to the dissipation of mechanical energy. Turbulent eddies of size A^> A 
(where A is the internal scale of the turbulence) lead to an equalisation of temperature by 
purely mechanical mixing of fluid particles which are at different temperatures. Consider- 
able true temperature gradients in regions of size A ~ A , on the other hand, are equalised 
by dissipative thermal conduction. 

The dissipation by thermal conduction (increase of entropy) is determined by the quantity 
x(grad TflT* (see (49.6)); supposing the turbulent fluctuations of temperature to be rela- 
tively small, we can replace T 2 in the denominator by a constant, the square of the mean 
temperature. According to the method described in §32 (see the first footnote to that section), 
we write Xturb (7a/A) 2 = constant. Substituting v t urb,A ~ »Wb,A ~ ~^x> *>A ~ (**)* (see 
(32.1)), we find the required relation to be T^ ~ A*. Thus for A^> A the temperature fluc- 
tuations, like the velocity fluctuations, are proportional to the cube root of the distance. 
At distances A <^ A , however, by the same arguments as for the velocity, the differences T^ 
are simply proportional to A. 

Problem 4. Derive a relation between the local correlation functions 



B TT = (T 2 - 7i)2, B iTT = (v 2i -v u )(T 2 - Ti)2 

in a non-uniformly heated turbulent flow (A. M. Yaglom 1949). 

Solution. The calculations are similar to those used in deriving formula (33.18). From 



t For the values of the thermal conductivity actually found, the Prandtl number does not reach 
the values for which this limiting law holds. Such laws can, however, be applied to convective diffu- 
sion; this obeys the same equations as convective heat transfer, but with the temperature replaced by 
the concentration of the solute, and the heat flux by the flux of solute, the "diffusion Prandtl number" 
being defined as Po = v/D, where D is the diffusion coefficient. For example, for solutions in water 
and similar liquids, Pj> reaches values of the order of 10 3 , while for very viscous solvents it is 10* or 
more. 

J The calculation of the constant in this formula for various particular cases is facilitated by the fact 
that, by virtue of the inequality S' <^ S, we need take only the first terms of an expansion, in powers of 
y, of the fluid velocity components in integrating equation (54.1) across the temperature boundary 
layer. Calculations for convective diffusion in various particular cases are given by V. G. Levich, 
Physico-chemical Hydrodynamics {Fiziko-khimicheskaya gidrodinamika) , Moscow 1952. 



§55 Heating of a body in a moving fluid 209 

the equations 

dT dT dvi n 

— + vi — = x&T, — = 
dt dxi oxi 



we find 



d 



-(7i T 2 ) = -2— (wT! T 2 ) + 2 x Ai(T 1 T 2 ). 

Ot OX\i 

On the left-hand side we put r = r 2 — r u and on the right we express the mean values in terms 
of the correlation functions, using the homogeneity and isotropy of the flow: 

a— l d 

— T 2 = --- — Bitt-xAiBtt- 
ot 2 ox\i 

Writing B t TT = «< B t tt and changing to derivatives with respect to r, we obtain an equation 
which, on integration over r, gives the required relation 

B r TT — 2xdBTTldr = —&<(>> 

where 



4> = -d(T*)\dt = -8(T-T) 2 ldt. 

Using the results of Problem 3, we then find that, for r^> A , B t tt = — ir<f>, while for 
r <^ A we have Btt = r 2 <f>/9x- 

§55. Heating of a body in a moving fluid 

A thermometer immersed in a fluid at rest indicates a temperature equal to 
that of the fluid. If the fluid is in motion, however, the thermometer indicates 
a somewhat higher temperature. This is because the fluid brought to rest 
at the surface of the thermometer is heated by internal friction. 

The general problem may be formulated as follows. A body of arbitrary 
shape is immersed in a moving fluid ; thermal equilibrium is established after 
a sufficient length of time, and it is required to determine the temperature 
difference T\ — To then existing between the body and the fluid. 

The solution of this problem is given by equation (50.2), in which, however, 
we cannot now neglect the term containing the viscosity as we did in (53.1); 
it is this term which is responsible for the effect under consideration. Thus 
we have for a steady state 

vgrad r = xA r + _(_! + -_± . (55.1) 

ZCp \ OXjc OX{ ] 

This must be supplemented by the equations of motion (53.3) of the fluid 
itself and also, strictly speaking, by the equation of thermal conduction in the 
body. In the limiting case where the body has a sufficiently small thermal 
conductivity, we can neglect the latter and suppose the temperature at any 
point on the surface of the body to be simply equal to the fluid temperature 
at that point, obtained by solving equation (55.1) with the boundary condition 



210 Thermal Conduction in Fluids §55 

dTjdn = 0, i.e. the condition that there is no heat flux through the surface of 
the body. In the opposite limiting case where the body has a sufficiently 
large thermal conductivity, we can use the approximate condition that the 
temperature should be the same at every point of its surface; the derivative 
dTjdn will not then in general vanish over the whole surface, and we must 
require only that the total heat flux through the surface of the body (i.e. the 
integral of BT/dn over the surface) should be zero. In both these limiting cases 
the thermal conductivity of the body does not appear explicitly in the solution 
of the problem, and we shall suppose in what follows that one of these cases 
holds.f 

Equations (55.1) and (53.3) contain the constant parameters x> v and c v , 
and their solutions involve also the dimension / of the body and the velocity 
U of the main stream. (The temperature difference Ti-T is not now an 
arbitrary parameter, but must itself be determined by solving the equations.) 
From these parameters we can construct two independent dimensionless 
quantities, which we take to be R and P. Then we can say that the required 
temperature difference 7i - To is equal to some quantity having the dimensions 
of temperature (which we take to be U 2 jc v ), multiplied by a function of R and 
P: 

r 1 -r = (^/c 3 ,)/(R,p). (55.2) 

It is easy to determine the form of this function for very small Reynolds 
numbers, i.e. for sufficiently small velocities U. In this case the term 
V'grad T in (55.1) is small compared with xAT, so that this equation be- 
comes 



XAT= - 



ZCp 






The temperature and velocity vary considerably over distances of the order 
of /. Hence an estimate of the two sides of equation (55.3) gives x(T\- T )/l 2 
~vU 2 /c p l 2 , or Ti-T ~ vU 2 Jxc p . Thus we conclude that, for small R, 

7i - To = constant x P U 2 /c p , (55.4) 

where the numerical constant depends on the shape of the body. It should 
be noticed that the temperature difference is proportional to the square of 
the velocity U. 

Some general conclusions concerning the form of the function /(P, R) in 
(55.2) can be drawn in the opposite limiting case of large R, when the velocity 
and the temperature vary only in a narrow boundary layer. Let 8 and S' be 
the distances over which the velocity and temperature respectively vary; 8 
and 8' differ by a factor depending on P. The amount of heat evolved in 
unit area of the boundary layer in unit time owing to the viscosity of the fluid 



t I. A. Kibel' has obtained an exact solution for the rotation of a heated disk in a viscous fluid, simi- 
lar to the solution given in §23 for a constant temperature; see Prikladnaya matematika i mekhanika 11, 
611, 1947. 



§55 Heating of a body in a moving fluid 211 

is the integral of \vp{dvijdx]c-\-dvjcjdxi) 2 over the thickness of the layer 
(see (16.3)). This integral is of the order of vp(U 2 jB 2 )8 — vpU 2 j8. The same 
amount of heat must be lost to the body, and it is therefore equal to the heat 
flux q =» — KdTjdn ~ x c vp{Ti — 7o)/o". Comparing the two expressions, we 
find 

T 1 -T = (U 2 lc p )f(P). (55.5) 

Thus, in this case, the function /is independent of R, but its dependence on 
P remains undetermined. 

PROBLEMS 

Problem 1. Determine the temperature distribution in a fluid moving in Poiseuille flow 
in a pipe of circular cross-section whose walls are maintained at a constant temperature T . 

Solution. In cylindrical co-ordinates, with the ar-axis along the axis of the pipe, we have 
v z — v = 2vm[l — (r/R)*\, where v m is the mean velocity of the flow. Substitution in (55.3) 
gives the equation 

1 d / dT\ \6v m 2 „ 

1 r I = r 2 . 

r dr \ dr J R* x c v 

The solution finite at r = and equal to T for r = R is 

r-r„ = ^[i-(-L)] 4 . 

Problem 2. Determine the temperature difference between a solid sphere and a fluid 
moving past it at small Reynolds numbers. The thermal conductivity of the sphere is 
supposed large. 

Solution. We take spherical co-ordinates r, 8, <f>, with the origin at the centre of the sphere 
and the polar axis in the direction of the velocity of the main stream. Calculating the com- 
ponents of the tensor dvildxk+dvk/dxi by means of formulae (15.17) and (20.9), we obtain 
equation (55.3) in the form 



1 d / „dT\ 1 d / 



1 -( 

r* 8r\ 



r z I + ( sin0- 

dr J r*smd dd\ 86/ 

= - A(Rlry [cos20{3 - 6(i?/r)2 + 2(i?/r)4} + (R/r)*] , 

where A = 9« a P/4c„. We look for T(r, 0) in the form T =f(r) cos 2 +g(r), and, separating 
the part which depends on 9, find two equations for /and g: 

r 2 f" + 2rf-6f = -A[3{R/r) 2 -6(R/ry+2(Rlr)% 
r 2 g" + Irg' + 2/ = - A(R/r)*. 

From the first we obtain 

/ = A[^R/r) 2 + (R/rY-MRIr) 6 ] + ci(Rlrf; 

the term of the form constant X r 2 is omitted, since it does not vanish at infinity. The second 
equation then gives 

g = -lAB(Rlr) 2 + i(RlrY+MRIr) 6 ]-MR/rf+c 2 Rlr+C3. 
The constants c u c 2 , c a are determined from the conditions 

T= constant and f (dT/dr)r* sin0d0 = 



212 Thermal Conduction in Fluids §56 

for r = R, which are equivalent to/(JR) = and g'(R)+\f'(R) = 0; also T = T at infinity. 
Thus c x = —5.4/3, c 2 = 2.4/3, c 8 = T . The temperature difference between T t = T(i?) 
and T is found to be T a — T = 5u 2 P/8c P . It may be noted that the temperature distribution 
obtained actually satisfies the condition dT/dr — for r = R, i.e. / '(R) = £'(.R) = 0. 
Hence it is also the solution of the same problem for a sphere of small thermal conductivity. 

§56. Free convection 

We have seen in §3 that, if there is mechanical equilibrium in a fluid in a 
gravitational field, the temperature distribution can depend only on the alti- 
tude z: T = T(z). If the temperature distribution does not satisfy this 
condition, but is a function of the other co-ordinates also, then mechanical 
equilibrium in the fluid is not possible. Furthermore, even if T = T(z), 
mechanical equilibrium may still be impossible if the vertical temperature 
gradient is directed downwards and its magnitude exceeds a certain value (§4). 

The absence of mechanical equilibrium results in the appearance of internal 
currents in the fluid, which tend to mix the fluid and bring it to a constant 
temperature. Such motion in a gravitational field is called free convection. 

Let us derive the equations describing this convection. We shall suppose 
the fluid incompressible. This means that the pressure is supposed to vary 
only slightly through the fluid, so that the density change due to changes in 
pressure may be neglected. For example, in the atmosphere, where the pres- 
sure varies with height, this assumption means that we shall not consider 
columns of air of great height, in which the density varies considerably over 
the height of the column. The density change due to the non-uniform heating 
of the fluid, of course, can not be neglected ; it results in the forces which 
bring about the convection. 

We write the variable temperature T(x, y, z, t) in the form T = To + T', 
where To is some constant mean temperature from which the variation T' is 
reckoned. We shall suppose that 7" is small compared with To. 

We write the fluid density also in the form p = p + p, with po a constant. 
Since the temperature variation 7" is small, the resulting density change p 
is also small, and we can write 

p' = (8pol8T) p T f = -pojSr. (56.1) 

Here /? = —(l/p)dpJ8T is the thermal-expansion coefficient of the fluid. 

In the pressure p = po+p', Po is not constant. It is the pressure cor- 
responding to mechanical equilibrium, when the temperature and density are 
constant and equal to To and po respectively. It varies with height according 
to the hydrostatic equation 

Po = Pog*^ + constant. (56.2) 

We start by transforming the Navier-Stokes equation, which has, in the 
presence of a gravitational field, the form 

8v/d* + (v»grad)v = -(l//>)gradp + vAv+g; 
this is obtained by adding the force g per unit mass to the right-hand side 



§56 Free convection 213 

of equation (15.7). We now substitute p = po+p\ p = po+p; to the first 
order of small quantities, we have 

grad^> grad/> gradp' grad^> 

— - — = + p ' t 

P po po po* 

or, substituting (56.1) and (56.2), 

grad/> grad/>' 

- - = g + - - + gT% 

P po 

With this expression, the Navier-Stokes equation gives 

Sv/^+(v.grad)v= -(l//>)grad/>' + vAv-j8rg, (56.3) 

where the suffix has been dropped from p . In the thermal conduction equa- 
tion (50.2), the viscosity term can be shown to be small in free convection 
compared with the other terms, and may therefore be omitted. We thus 
obtain 

dT'/dt+vgrad T = X AT*. (56.4) 

Equations (56.3) and (56.4), together with the equation of continuity 
div v = 0, form a complete system of equations governing free convection. 
For steady flow, the equations of convection become 

(v.grad)v= -(l//>)grad£'-j8rg + vAv, (56.5) 

v.gradT' = X AT', (56.6) 

divv = 0. (56.7) 

This system of five equations for the unknown functions v, p'[p and V 
contains three parameters, v, x and fig. Moreover, the solution will involve 
a characteristic length / and the temperature difference Ti - T between the 
solid body and the fluid at a great distance. There is here no characteristic 
velocity, since there is no flow due to external forces, and the whole motion 
of the fluid is due to its non-uniform heating. 

Thus steady free convection in a gravitational field is characterised by 
five parameters, which have the following dimensions: x = v = cm 2 /sec, 
Ti —Tq = deg, / = cm, fig = cm/sec 2 deg. From these we can form two 
independent dimensionless quantities, which we take to be the Prandtl 
number P = v/x and the Grashof number 

G = ^(T 1 -T )I^. (56.8) 

The similarity law for free convection is therefore 

v = (v//)f(r//, G), T = (Ti - 7o)/(r//, P, G). (56.9) 

Two flows are similar if their Prandtl and Grashof numbers are the same. 
Convective heat transfer caused by gravity is again characterised by the 



214 Thermal Conduction in Fluids §56 

Nusselt number, which is now a function of P and G only: 

N=/(P,G). (56.10) 

The value of the Grashof number is an important characteristic of con- 
vective flow. When G is sufficiently small, the free convection is unimportant 
in the heat transfer in the fluid, which is then due mainly to ordinary con- 
duction. 

Convective flow may be either laminar or turbulent. There is no Reynolds 
number for free convection (since there is no characteristic velocity para- 
meter), and the onset of turbulence is determined by the Grashof number : 
the convection becomes turbulent when G is very large. 

A very curious case of convection is the flow which occurs in a fluid between 
two infinite horizontal planes at different temperatures, that of the lower plane 
(T2) being greater than that of the upper plane (Ti). If the temperature 
difference T% — T\ is small, the fluid remains at rest and there is pure thermal 
conduction, the fluid temperature and density being functions only of the 
vertical co-ordinate z; the density increases upward. If the difference T2—T1 
exceeds a certain critical value, however, which depends on the distance / 
between the planes, such a state becomes unstable and steady convection 
occurs. The onset of instability can be determined theoretically (see Problem 
5). The critical value of the difference T%— T\ appears as a factor in the 
product 

GP = ^Z3(r 2 -Ti)/v X . (56.11) 

In a layer of fluid between two solid planes at constant temperatures, con- 
vection must occur if GP > 1710. If the upper surface is free, but still at a 
constant temperature, then convection occurs for GP > HOO.f 

The convective flow which occurs is somewhat unusual. Since the fluid is 
unbounded in the horizontal plane, it is evident that the flow must be periodic 
in that plane. In other words, the space between the bounding planes must be 
divided into similar right prisms in each of which the fluid moves in a similar 
way. The horizontal cross-sections of these prisms form a network in the 
horizontal plane. The theoretical determination of the nature of this network 
is very difficult, but experimental results seem to indicate that there is a 
hexagonal pattern with cells in the form of hexagonal prisms, the fluid moving 
up in the middle and down at the edges, or else vice versa. 

For very large values of G, the steady convection in turn becomes unstable ; 
turbulence sets in for G ~ 50,000. 

Another similar case of instability is that of convection in a vertical 
cylindrical pipe along which a constant temperature gradient is maintained. 



t These conditions (for a given difference T 2 — T t ) are always fulfilled if / is sufficiently large. To 
avoid misunderstanding, we should mention that we are speaking here of values of / for which the 
variation in the fluid density under the action of gravity is unimportant. Hence the above criteria 
cannot.be applied to gas columns of great height. In this case we have to use the criterion derived in 
§4, from which we see that convection need not occur for a column of any height if the temperature 
gradient is small enough. 



§56 Free convection 215 

Here again there is a critical value of the product GP beyond which the fluid 
at rest is unstable; see Problem 6. 

PROBLEMS 

Problem 1. Determine the Nusselt number for free convection on a flat vertical plate. 
It is assumed that the velocity and the temperature difference T = T—T (where T is 
the fluid temperature at infinity) are appreciably different from zero only in a thin boundary 
layer adjoining the surface of the plate (K. PohlhaUsen). 

Solution. We take the origin on the lower edge of the plate, the *-axis vertical, and the 
y-axis perpendicular to the plate. The pressure in the boundary layer does not vary along the 
}>-axis (cf. §39), and therefore is everywhere equal to the hydrostatic pressure p (x), i.e. 
p' = 0. With the usual accuracy of boundary-layer theory, equations (56.5)-(56.7) become 

a y =^ + ^-r„), (i) 



BT BT B*T 

. \- Vy = ^ — 

Bx By By 



+ *«r£: = xzi. ( 2 ) 



Bv x Bv y 

with the boundary conditions v x = v y = and T = T± for y = (T x being the temperature 
of the plate), v x = and T = T for y = oo. These equations can be converted into ordinary 
differential equations by introducing as the independent variable 

€ = Cyjx\ C = \MTi- ?o)/4v2]*. (4) 

We put 

v x = AvCW^'^l T-T = (T 1 - r o )0(£). (5) 

Then (3) gives v y = vCx-h({<f>'-3<f>), and (1) and (2) give equations for <f> and 0: 

41" + 3cfxf>"-24'2 + d = 0, 6" + 3P<£0' = 0, (6) 

with the boundary conditions #0) = f (0) = 0, 6(0) = 1, f(oo) = 0, «(oo) = 0. It follows 
from (4) and (5) that the thickness of the boundary layer is of the order S ~ x i JC. The con- 
dition for the solution to be valid is therefore S < / (where I is the height of the plate), or 
G* ^> 1. The total heat flux per unit area of the plate is 







The Nusselt number is N =/(P)G*, where the function /(P) is determined by solving the 
equations (6). 

Problem 2. A hot turbulent submerged jet of gas is bent round by a gravitational field : find 
its shape (G. N. Abramovich 1938). 

Solution. Let T be some mean value (over the cross-section of the jet) of the temperature 
difference between the jet and the surrounding gas, u some mean velocity of the gas in the 
jet, and / the distance along the jet from its point of entry; I is supposed large compared with 
the dimensions of the aperture by which the jet enters. The condition of constant heat flux 
Q along the jet is Q ~ pc^TuR^- = constant and, since the radius of a turbulent jet is pro- 
portional to / (cf. §35), we have 

T'ul 2 = constant ~ Q/pCp] (1) 



216 Thermal Conduction in Fluids §56 

we notice that, in the absence of the gravitational field, u f* \jl (see (35.3)) and it then follows 
from (1) that T ~ 1/Z. 

The momentum flux vector through the cross-section of the jet is proportional to pu 2 R 2 n 
~ gu 2 l 2 n, where n is a unit vector along the jet. Its horizontal component is constant along 
the jet: 

m 2 / 2 cos 6 = constant, (2) 

id the horizontal, while the chang 
et. This force is proportional to 

ppgT'R* ~ ppgT'l* ~ feQlcp. 



where 8 is the angle between n and the horizontal, while the change in the vertical component 
is due to the "lift force" on the jet. This force is proportional to 



Hence we have 

d(Z 2 w 2 sin 0)/dZ - PgQIpcpU. (3) 

It then follows from (2) that d(tan 0)/dl = constant X / cos* 0, whence we obtain finally 





/ 



dd 

= constant x I 2 , (4) 



cos 5/2 



where O gives the direction of the emergent jet. 

In particular, if does not vary appreciably along the jet, (4) gives 0— O = constant X/ 2 . 
This means that the jet is a cubical parabola, in which the deviation d from a straight line is 
d = constant X I 3 . 

Problem 3. A turbulent jet of heated gas (i.e. one with a large Grashof number) rises from 
a fixed hot body. Determine the variation of the velocity and temperature in the jet with 
height (Ya. B. Zel'dovich 1937). 

Solution. As in the preceding case, the radius of the jet is proportional to the distance 
from its source, and we have, analogously to (1) of Problem 2, T'uz 2 = constant, and instead 
of (3) d(z 2 u 2 )Jdz = constant/w, where z is the height above the body, supposed large compared 
with the dimension of the body. Integrating, we find u ~ z~*, and for the temperature 

r ~z- s '\ 

Problem 4. The same as Problem 3, but for a laminar convective jet rising freely (Ya. 
B. Zel'dovich 1937). 

Solution. Together with the relation T'uR 2 = constant, which expresses the constancy 
of the heat flux, we have u 2 fz ~ vu/R 2 ~ PgT', which follows from equation (56.5). From 
these relations we find the following variation of the radius, velocity and temperature with 
height: R ~ ^z, u = constant, T ~ 1/z. It may be noticed that the number G ~ T'R 3 
~ \/z, i.e. increases with height, and the jet must therefore become turbulent at a certain 
altitude. 

Problem 5. Derive the equations governing the onset of steady convection between two 
horizontal planes maintained at given temperatures (Rayleigh 1916). 

Solution. A perturbation proportional to e~ iu>t is applied to a fluid at rest with a constant 
vertical temperature gradient dT/dz = —A < 0. The state of rest is unstable if there is any 
possible value of to whose imaginary part is positive. Hence the onset of instability is deter- 
mined by the appearance of a solution for which the imaginary part of w is zero. In this case 
we are concerned with the appearance of steady convection as a result of instability; hence 
we must seek solutions for which the real part of w is also zero, that is, solutions independent 
of time. 

In equations (56.5)-(56.7), the velocity v of the perturbing motion and the resulting pressure 
variation p' are small quantities. We write the temperature as T = —Az+r, where the 



§56 Free convection 217 

perturbation t is small ; we suppose the pressure variation resulting from the constant tem- 
perature gradient to be included in p . Then we find, omitting second-order terms, 

vAv = gjcad(p'lp)+pTg, 
xAr = -Av Zt divv = 0. * ' 

Eliminating v and p'/p, we obtain an equation for t: 

where y = V$gA\vx — GP, and Z is the distance between the planes. 

The boundary conditions on equations (1) at a solid surface are t = 0, v z — 0, 8v z \8z — 0. 
The last of these follows from the equation of continuity, since we must have v x = v y = 
for all x and y. By the second equation (1), the conditions on v z can be replaced by conditions 
on higher derivatives of t, c 2 being replaced by 8 2 rj8z 2 . 

We look for t in the form e ik ' T f(z), where k is a vector in the xy-plane, and obtain for f(z) 
the equation 

d 2 \3 yk 2 



1 d* \ 3 y& 



The general solution of this equation is a linear combination of the functions cosh((iz/l) 
and sinh^z/l), where n 2 = k 2 l 2 —y*(klfty\ with the three different values of ^/\. The 
coefficients are determined by the boundary conditions, which lead to a system of algebraic 
equations; the compatibility condition then determines the function kl(y). The inverse 
function y = y(kl) has a minimum for some value of kl; the corresponding y = GP deter- 
mines the required criterion for the appearance of instability, and the value of k determines 
the periodicity in the xy-plane, but not the symmetry, of the resulting motion, f 

Problem 6. Determine the onset of steady convection in a fluid at rest in a vertical cylin- 
drical pipe along which a constant temperature gradient is maintained (G. A. Ostroumov 
1946). 

Solution. We seek a solution of the equations (1) of Problem 5 in which the convective 
velocity v is everywhere parallel to the axis of the pipe (the .s-axis), and the flow pattern does 
not vary along this axis, i.e. v z = v, r and dp'ldz depend only on the co-ordinates xand y. 
Then the equations become Bp'Jdx = 0, dp' I By = 0, vA 2 v = —PgT+(l[ P )dp'ldz, xA 8 t 
= — Av, where A 2 = d 2 /8x 2 + 8 2 j8y 2 . The first two equations show that dp'fdz = constant, 
and, eliminating t from the other equations, we have 

where we have again put y = AR^g/xv = GP, and R is the radius of the pipe. At the surface 
of the pipe we must have v = and the heat flux continuous. Moreover, the total mass flux 
through a cross-section of the pipe must be zero. 

Equation (1) has solutions of the form Jn(kr) cos rt(f> and I n (kr) cos n<f>, where J n and I n 
are Bessel functions of real and imaginary argument respectively, r and <j> are polar co-ordi- 
nates in the cross-section, and kR = y*. The onset of convection corresponds to the solution 
for which y is least. It is found that this is the solution with n = 1 : 

v = vocostUiikry^kty-hikrViikR)], 
r = voiv&l^costtMkrWkiq + hikrViikR)]. 



f A detailed account of the calculations is given by A. Pellew and R. V. Southwell, Proceedings 
of the Royal Society A176, 312, 1940. 



218 Thermal Conduction in Fluids §56 

The pressure gradient dp'Jdz does not appear. The condition v = for r — R is satisfied 
identically, and the total mass flux through the cross-section of the pipe is zero. In the 
limiting case of thermally insulating walls, we must have also drjdr = for r = R, or 

JojkR) IojkR) _ 2 
j!(kR) + h(kR) kR 

The smallest root of this equation gives the required critical value of y = (kR)* = 67-4. 
In the opposite limiting case of walls of infinite thermal conductivity, we must have t = 
for r = R; thenJi(kR) = 0, whence the critical value is y = 215*8.f 



t For a more detailed discussion see G. A. Ostroumov, Free Convection in a Confined Medium 
(Svobodnaya konvektsiya v usloviyakh vnutrennei zadachi), Moscow 1952. 



CHAPTER VI 

DIFFUSION 

§57. The equations of fluid dynamics for a mixture of fluids 

Throughout the above discussion it has been assumed that the fluid is 
completely homogeneous. If we are concerned with a mixture of fluids 
whose composition is different at different points, then the equations of 
fluid dynamics are considerably modified. 

We shall discuss here only mixtures with two components. The com- 
position of the mixture is described by the concentration c, defined as the 
ratio of the mass of one component to the total mass of the fluid in a given 
volume element. 

In the course of time, the distribution of the concentration through the 
fluid will in general change. This change occurs in two ways. Firstly, when 
there is macroscopic motion of the fluid, any given small portion of it moves 
as a whole, its composition remaining unchanged. This results in a purely 
mechanical mixing of the fluid; although the composition of each moving 
portion of it is unchanged, the concentration of the fluid at any point in space 
varies with time. If we ignore any processes of thermal conduction and inter- 
nal friction which may also be taking place, this change in concentration is a 
thermodynamically reversible process, and does not result in the dissipation 
of energy. 

Secondly, a change in composition can occur by the molecular transfer of 
the components from one part of the fluid to another. The equalisation of the 
concentration by this direct change of composition of every small portion of 
fluid is called diffusion. Diffusion is an irreversible process, and is, like 
thermal conduction and viscosity, one of the sources of energy dissipation in a 
mixture of fluids. 

We denote by p the total density of the fluid. The equation of continuity 
for the total mass of the fluid is, as before, 

8 P l8t + div(pv) = 0. (57.1) 

It signifies that the total mass of fluid in any volume can vary only by the 
movement of fluid into or out of that volume. It must be emphasised that, 
strictly speaking, the concept of velocity itself must be redefined for a mixture 
of fluids. By writing the equation of continuity in the form (57.1), we have 
defined the velocity, as before, as the total momentum of unit mass of fluid. 

The Navier-Stokes equation (15.5) is also unchanged. We shall now derive 
the remaining equations of fluid dynamics for a mixture of fluids. 

In the absence of diffusion, the composition of any given fluid element 
would remain unchanged as it moved about. This means that the total 

219 



220 Diffusion §57 

derivative dcjdt would be zero, i.e. the equation dc/dt = dc/dt+v-grad c = 
would hold. This equation can be written, using (57.1), as 

d(pc)ldt + div(pcv) = 0, 

i.e. as an equation of continuity for one of the components of the mixture 
(pc being the mass of that component in unit volume). In the integral form 



— pcdV = — (b pcv-df 



it shows that the rate of change of the amount of this component in any 
volume is equal to the amount of the component transported through the 
surface of that volume by the motion of the fluid. 

When diffusion occurs, besides the flux pcv of the component in question 
as it moves with the fluid, there is another flux which results in the transfer 
of the components even when the fluid as a whole is at rest. Let i be the 
density of this diffusion flux, i.e. the amount of the component transported 
by diffusion through unit area in unit time.f Then we have for the rate of 
change of the amount of the component in any volume 

— pcdV = — (ppcv'df— (|)i»df, 

or, in differential form, 

d{pc)\dt = - div(pcv) - div i. (57.2) 

Using (57.1), we can rewrite this "equation of continuity" for one component 
in the form 

p(dcjdt+v grade) = -divi. (57.3) 

To derive another equation, we repeat the arguments given in §49, bearing 
in mind that the thermodynamic quantities for the fluid are now functions of 
the concentration also. In calculating the derivative 8(%pv 2 + pe)jdt (in §49) 
by means of the equations of motion, we had to transform the terms pdejdt 
and — v-gradp. This transformation must now be modified, because the 
thermodynamic identities for the energy and the heat function now contain an 
additional term involving the differential of the concentration: 

dc = Tds+(plpZ)dp + fj,dc, 
dw = Tds + (l[p)dp+fidc, 



f The sum of the flux densities for the two components must be pv. If the flux density for one 
component is pev+i, that for the other component is therefore p(l — c)v— i. 



§57 The equations of fluid dynamics for a mixture of fluids 221 

where p. is an appropriately denned chemical potential of the mixture.f 
Accordingly, an additional term p/idc/dt appears in the derivative pde/dt. 
Writing the second thermodynamic relation in the form 

dp = pdw—pTds—pfxdc, 

we see that the term — vgradp will contain an additional term ppv -grad c. 
Thus we must add pp{dcJdt + \- grade) to the expression (49.3). By 
equation (57.3), this can be written -p. div i. The result is 

d 

—(ipv 2 +pe) = — div\pv(lv 2 + w)— v»o' + q] + 
8t 

(8s \ dvt 
+pT\ — + vgrads) - a'ik h divq-ju, divi. (57.4) 

\ dt 1 dxic 

We have replaced - k grad T by a heat flux q, which may depend not only 
on the temperature gradient but also on the concentration gradient (see the 
next section). The sum of the last two terms on the right can be written 

divq— jLtdivi = div(q— jui)+i-grad^. 

The expression p\(%v 2 + w) — v«a' + q which is the operand of the diver- 
gence operator in (57.4) is, by the definition of q, the total energy flux in 
the fluid. The first term is the reversible energy flux, due simply to the 
movement of the fluid as a whole, while the sum — vo' + q is the irreversible 
flux. When there is no macroscopic motion, the viscosity flux v«o' is zero, 
and the thermal flux is simply q. 

The equation of conservation of energy is 

8 
—Upv 2 +pe)= -div[pv(|t; 2 + ro)-v.o' + q]. (57.5) 

Subtracting from (57.4), we obtain the required equation 
8s _ , \ , 3©< 

8xjc 
which is a generalisation of (49.4). 



(8s \ dvt 

P T \ ~Z + v 'S rad * = a ' ik ~o div(q-ju)-i.grad/A, (57.6) 

\ ot J dxic 



t It is known from thermodynamics that, for a mixture of two substances, the thermodynamic 
identity is 

de = Tds— pdV+fiidni+p,2dn2, 

where n lt n^ are the numbers of particles of the two substances in 1 g of the mixture, and (j. lt ^ are 
the chemical potentials of the substances. The numbers n v n^ satisfy the relation « 1 OT 1 +n 2 OT 2 = 1, 
where m 1 and n^ are the masses of the two kinds of particle. If we introduce as a variable the 
concentration c = n^m x , we have 

de= Tds-pdV+(^-^)dc. 
\m\ mil 

Comparing this with the relation given in the text, we see that the chemical potential fi is related 
to fa and /ig by 

_ pi H>2 
mi 7»2 



222 Diffusion §58 

We have thus obtained a complete system of equations of fluid mechanics 
for a mixture of fluids. The number of equations in this system is one more 
than for a single fluid, since there is one more unknown function, namely the 
concentration. The equations are the equation of continuity (57.1), the 
Navier-Stokes equations, the "equation of continuity" (57.2) for one com- 
ponent, and equation (57.6), which determines the change in entropy. 
It must be noticed that equations (57.2) and (57.6) as they stand determine 
only the form of the corresponding equations of fluid dynamics, since they 
involve the undetermined fluxes i and q. These equations become determi- 
nate only when i and q are replaced by expressions in terms of the gradients 
of concentration and temperature. The corresponding expressions will be 
obtained in §58. 

For the rate of change of the total entropy of the fluid, a calculation entirely 
similar to that of §49, but using (57.6) in place of (49.4), gives the result 

1 J„dF - - J ^yV J^dr + ..., (57.7) 

where we have omitted, for brevity, the viscosity terms. 

§58. Coefficients of mass transfer and thermal diffusion 

The diffusion flux i and the heat flux q are due to the presence of con- 
centration and temperature gradients in the fluid. It should not be thought, 
however, that i depends only on the concentration gradient and q only on the 
temperature gradient. On the contrary, each of these fluxes depends, in 
general, on both gradients. 

If the concentration and temperature gradients are small, we can suppose 
that i and q are linear functions of grad fi and grad T.-f Accordingly, 
we write i and q as 

i= -agrad/A-jSgradT, q= -S grad [M-y grad T+fxi. 

There is a simple relation between the coefficients /? and 8, which is a 
consequence of a symmetry principle for the kinetic coefficients. This symmetry 
principle is as follows.! 

Let us consider some closed system, and let xi, X2, ... be some quantities 
characterising the state of the system. Their equilibrium values are deter- 
mined by the fact that, in statistical equilibrium, the entropy S of the whole 
system must be a maximum, i.e. we must have X a = for all a, where X a 
denotes the derivative 

X a = -dSldx a . (58.1) 

We assume that the system is in a state near to equilibrium. This means that 



t The fluxes q and i are independent of the pressure gradient (for given grad fx and grad T), for 
the same reason as that given with regard to q in §49. 

X See Statistical Physics, §119, Pergamon Press, London 1958. 



§58 Coefficients of mass transfer and thermal diffusion 223 

all the x a are very little different from their equilibrium values, and the 
X a are small. Processes will occur in the system which tend to bring it into 
equilibrium. The quantities x a are functions of time, and their rate of change 
is given by the time derivatives x a ; we express the latter as functions of X a , 
and expand these functions in series. As far as terms of the first order we have 

x a = - ^2yabX b . (58.2) 

b 

The symmetry principle for the kinetic coefficients states that the y ab (called 
the kinetic coefficients) are symmetrical with respect to the suffixes a and b : 

Yab = 7ba- (58.3) 

The rate of change of the entropy S is 

S= -T,X a x a . (58.4) 

Now let the x a themselves be different at different points of the system, i.e. 
each volume element have its own values of the x a . That is, we suppose the x a 
to be functions of the co-ordinates. Then, in the expression for S, besides 
summing over a we must integrate over the volume of the system: 

S = - (^X a x a dV. (58.4a) 

J a 

It is usually true that the values of the x a at any given point depend only on 
the values of the X a at that point. In this case we can write down the 
relation between x a and X a for each point in the system, and obtain the same 
formulae as previously.f 

In the problem under consideration we take as the x a the components of 
the vectors i and q— [A. Then we see from a comparison of (57.7) and (58.4a) 
that the X a are respectively the components of the vectors (l/T)grad/j 
and (1/T 2 ) grad T. The kinetic coefficients y a b are the coefficients of these 
vectors in the equations 

i--^(!^)-/^J=JI). 
q -,i=-ar(^)- y r*(i^). 

By the symmetry of the kinetic coefficients, we must have /ST 2 = 8T, or 
8 = (3T. This is the required relation. 



t Strictly speaking, in order to apply the relations obtained for a discrete set of quantities to a 
continuous distribution, we should write the integral (58.4a) as a sum over small but finite 
regions AV of the body (cf. §132); then the definition of the coefficients yab also involves AV. In 
the present case, however, this procedure is unnecessary, since we use only the symmetry of the 
kinetic coefficients, and not their actual values. 



224 Diffusion §58 

We can therefore write the fluxes i and q as 

i = -a grad/*-£ grad T, 

q = _j8r grad/*-y grad T+/>ti, 

with only three independent coefficients a, /?, y. It is convenient to eliminate 
grad ju from the expression for the heat flux, replacing it by i and grad T. 
Then we have 

i = -agrad^-jSgradr, (58.6) 

q = (^+j8T/a)i- k grad T, (58.7) 
where 

K = y-p2T/oL. (58.8) 

If the diffusion flux i is zero, we have pure thermal conduction. For this 
to be so, T and p must satisfy the equation a grad ju + fi grad T = 0, or 
adj^+^dT = 0. The integration of this equation gives a relation of the 
form/(c, T) — which does not contain the co-ordinates explicitly. (The 
chemical potential is a function of the pressure, as well as of c and T, but in 
equilibrium the pressure is constant.) This relation determines the depen- 
dence of the concentration on the temperature which must hold if there is 
no diffusion flux. Moreover, for i = we have from (58.7) 

q = — k grad T, 

so that k is just the thermal conductivity. 

Let us now change to the usual variables p, T and c. We have 

grad/z = (dpi dc) v . T grad c + {d[xj '8T) C>P grad T+ (dfi/ dp) cT gradp. 

In the last term we can replace the derivative (dpjdp) Ct T by (dVjdc) Pt T, 
where V is the specific volume.f Substituting in (58.6) and (58.7), and putting 



p \ dc / T,p 



(58.9) 



P k T DIT = v.(dpjdT) CtV +p y 
k P = p(dVldc) PiT l(dpldc) p T , (58.10) 

we obtain 

i = - P D[gradc+(k T IT) grad T+(k p /p) grad/>], (58.11) 

q = [k T (Spl8c) p , T - T(8pldT) p , c +p]i- k grad T. (58.12) 

The coefficient D is called the diffusion coefficient or mass transfer coefficient] 



f The equality of these two derivatives follows from the thermodynamic identity 

d<f> = -sdT+Vdp + pdc, 

where <f> is the thermodynamic potential per unit mass; 

(dpjdp) c>T = dm dp 8c = (8VI8c) PfT . 



§58 Coefficients of mass transfer and thermal diffusion 225 

it gives the diffusion flux when only a concentration gradient is present. 
The diffusion flux due to the temperature gradient is given by the thermal 
diffusion coefficient UtD\ the dimensionless quantity Ay is called the thermal 
diffusion ratio. 

The last term in (58.11) need be taken into account only when there is a 
considerable pressure gradient in the fluid (caused by an external field, say). 
The coefficient kpD may be called the barodiffusion coefficient. It should 
be noticed that, by formula (58.10), the dimensionless quantity k p is entirely 
determined by thermodynamic properties alone. 

In a single fluid there is, of course, no diffusion flux. Hence it is clear that 
kr and k p must vanish in each of the two limiting cases c = and c = 1. 

The condition that the entropy must increase places certain restrictions on 
the coefficients in formulae (58.6) and (58.7). Substituting these formulae 
in the expression (57.7) for the rate of change of the entropy, we find 

8 r f /c(gradT) 2 C i 2 , „ n *„ 

Hence it is clear that, besides the condition k > which we already know, 
we must have also a > 0. Bearing in mind that the derivative (d[Mldc) Pt T 
is always positive,f we therefore find that the diffusion coefficient must be 
positive : D > 0. The quantities &t and k p , however, may be either positive 
or negative. 

We shall not pause to write out the lengthy general equations obtained by 
substituting the above expressions for i and q in (57.3) and (57.6). We 
shall take only the case where there is no significant pressure gradient, while 
the concentration and temperature of the fluid vary so little that the coeffi- 
cients in the expressions (58.11) and (58.12) may be supposed constant, 
although they are in general functions of c and T. Furthermore, we shall 
suppose that there is no macroscopic motion in the fluid except that which 
may be caused by the temperature and concentration gradients. The velocity 
of this motion is proportional to the gradients, and the terms in equations 
(57.3) and (57.6) which involve the velocity are therefore quantities of the 
second order, and may be neglected. The term — i«grad y. in (57.6) is also of 
the second order. Thus we have pdcjdt + div i = 0, pTds/dt + div(q— fii) = 0. 

Substituting for i and q the expressions (58.11) and (58.12) (without the 
term in gradp), and transforming the derivative dsjdt as follows: J 

8s I ds \ 8T I 8s \ 8c c p 8T I 8\i \ 8c 
Yt~ \~8f) CtV ~8t \~8c ) T ,p^t ~ ~f ~8t ~ \~8T ) Pte 8t 



t See Statistical Physics, §95. 
j For 



{8sj8c) PtT = -d^/dcST = -(dfj,l8T) p>c . 



226 Diffusion §58 

we obtain after a simple calculation 

dc/dt = D[Ac+(k T /T)ATl (58.14) 

dT/dt - (k T /c p )(dfM/dc) PiT 8c/dt = x A 7 1 . (58.15) 

This system of linear equations determines the temperature and concentra- 
tion distributions in the fluid. 

There is a particularly important case where the concentration is small. 
When the concentration tends to zero, the diffusion coefficient tends to a 
finite constant, but the thermal diffusion coefficient tends to zero. Hence 
kr is small for small concentrations, and we can neglect the term krA T 
in (58.14), which then becomes the diffusion equation 

dc/dt = DAc- (58.16) 

The boundary conditions on the solution of (58.16) are different in 
different cases. At the surface of a body insoluble in the fluid the normal 
component of the diffusion flux i = — pD grad c must vanish, i.e. we 
must have dc/8n = 0. If, however, there is diffusion from a body which 
dissolves in the fluid, equilibrium is rapidly established near its surface, 
and the concentration in the fluid adjoining the body is the saturation 
concentration Co', the diffusion out of this layer takes place more slowly 
than the process of solution. The boundary condition at such a surface is 
therefore c = cq. Finally, if a solid surface absorbs the diffusing substance 
incident on it, the boundary condition is c — 0; an example of such a case is 
found in the study of chemical reactions at the surface of a solid. 

Since the equations of pure diffusion (58.16) and of thermal conduction 
(50.4) are of exactly the same form, we can immediately apply all the formulae 
derived in §§51 and 52 to the case of diffusion, simply replacing T by c and 
X by D. The boundary condition for a thermally insulating surface corres- 
ponds to that for an insoluble surface, while a surface maintained at a constant 
temperature corresponds to a soluble surface from which diffusion takes place. 
In particular, we can write down, by analogy with (51.6), the following 
solution of the diffusion equation : 

M 

C{r) = J^Dif tM ~ r2lm ' (58 - 17) 

This gives the distribution of the solute at any time, if at time t = it is 
all concentrated at the origin (M being the total amount of the solute). 

PROBLEM 

Determine the barodiffusion coefficient for a mixture of two perfect gases. 
Solution. We have for the specific volume V = kT^+n^Jp (the notation is that used 
in the second footnote to §57), and the chemical potentials aret 

j"l = fl{p, T) + kT log[m/(»i + W 2 )], 

/*2 = flip, r) + ^riog[w 2 /(wi + « 2 )]. 



f See Statistical Physics, §92. 



§59 Diffusion of particles suspended in a fluid 227 

The numbers n x and n 2 are expressed in terms of the concentration of the first component by 
n imi = c, n 2 ms — 1 —c. A calculation using formula (58.10) gives 



rl-c C 1 

k p = (m2-mi)c(l-c)\ + — . 

L «*2 mi] 



§59. Diffusion of particles suspended in a fluid 

Under the influence of the molecular motion in a fluid, particles suspended 
in the fluid move in an irregular manner (called the Brownian motion). 
Let one such particle be at the origin at the initial instant. Its subsequent 
motion may be regarded as a diffusion, in which the concentration is repre- 
sented by the probability of finding the particle in any particular volume 
element. To determine this probability, therefore, we can use the solution 
(58.17) of the diffusion equation. The possibility of this procedure is due to 
the fact that, for diffusion in weak solutions (i.e. when c <4 1, which is when 
the diffusion equation can be used in the form (58.16)), the particles of the 
solute hardly affect one another, and so the motion of each particle can be 
considered independently. 

Let w(r, t)6r be the probability of finding the particle at a distance between 
r and r + dr from the origin at time t. Putting in (58.17) Mjp = 1 and 
multiplying by the volume 47rr 2 dr of the spherical shell, we find 

w ( r > W r = ^tWn ex P( " r ^ Dt) r2 dr ' (59>1) 

Let us determine the mean square distance from the origin at time t. 
We have 

00 

r 2 = j r ^w(r,t)dr. (59.2) 

o 

The result, using (59.1), is 

^ = 6Dt. (59.3) 

Thus the mean distance travelled by the particle during any time is propor- 
tional to the square root of the time. 

The diffusion coefficient for particles suspended in a fluid can be cal- 
culated from what is called their mobility. Let us suppose that some constant 
external force f (the force of gravity, for example) acts on the particles. In a 
steady state, the force acting on each particle must be balanced by the drag 
force exerted by the fluid on a moving particle. When the velocity is small, 
the drag force is proportional to it and is v[b, say, where b is a constant. 
Equating this to the external force f, we have 

v = bt, (59.4) 

i.e. the velocity acquired by the particle under the action of the external force 



228 Diffusion §59 

is proportional to that force. The constant b is called the mobility, and can 
in principle be calculated from the equations of fluid dynamics. For example, 
for spherical particles of radius R, the drag force is 6ttt)Rv (see (20.14)), 
and therefore the mobility is 

b = 1I6tt7]R. (59.5) 

For non-spherical particles, the drag depends on the direction of motion; 
it can be written in the form aacVjc, where aw is a symmetrical tensor (see 
(20.15)). To calculate the mobility we have to average over all orientations 
of the particle; if a\, az, a% are the principal values of the symmetrical tensor 
ciik, then we have 

b = - — + — + —. (59.6) 

3\ai a 2 as/ 

The mobility b is simply related to the diffusion coefficient D. To derive 
this relation, we write down the diffusion flux i, which contains the usual 
term — pD grad c due to the concentration gradient (we suppose the tem- 
perature constant), and also a term involving the velocity acquired by the 
particle owing to the external forces. This latter term is evidently pcv. 
Thus 

i = - P D grad c + P cM y (59.7) 

where we have used the expression (59.4). In a state of thermodynamic 
equilibrium, there is no diffusion, and the flux i must be zero. The equili- 
brium distribution of the concentration of particles suspended in a fluid, 
in an external field, is determined by Boltzmann's formula, according to 
which c = constant xr p/w , U being the potential energy of the particle 
in the external field. Since f = — grad U, we find the equilibrium concen- 
tration gradient to be grad c = cf/kT. Substituting this in (59.7) and equat- 
ing i to zero, we have 

D = kTb. (59.8) 

This is Einstein's relation between the diffusion coefficient and the mobility. 
Substituting (59.5) in (59.8), we find the following expression for the 
diffusion coefficient for spherical particles: 

D = kT/67T7]R. (59.9) 

Besides the translatory Brownian motion and diffusion of suspended par- 
ticles, we may consider also their rotary Brownian motion and diffusion. Just 
as the translatory diffusion coefficient is calculated in terms of the drag 
force, so the rotary diffusion coefficient can be expressed in terms of the 
forces on a particle executing a rotary movement in the fluid.f 



t If (non-spherical) particles are suspended in a plane-parallel stream with a transverse velocity 
gradient, a definite distribution of the particles as regards their orientation in space is established as 
a result of the simultaneous action of the orienting forces of fluid dynamics and the disorienting 
Brownian motion. For the solution of this problem for ellipsoidal particles, see A. Peterlin and H. A. 
Stuart, Zeitschrift fur Physik 112, 1, 1939. 



§59 Diffusion of particles suspended in a fluid 229 

PROBLEMS 

Problem 1. Particles execute Brownian motion in a fluid bounded on one side by a plane 
wall; particles incident on the wall "adhere" to it. Determine the probability that a particle 
which is at a distance x Q from the wall at time t = will have "adhered" to it after a time t. 

Solution. The probability distribution zo(x, t) (where x is the distance from the wall) 
is determined by the diffusion equation, with the boundary condition w = for x = 
and the initial condition w = §(x—x ) for t = 0. Such a solution is given by formula (52.4) 
when T is replaced by vi, x by D, and T (x') in the integrand by S(*'— x ). We then obtain 

«<*>') = -TTT^- zrT{^p[-(x-xofl4Dt]-exp[-(x+xofl4Dt]}. 
2s/\TTDt) 

The probability of "adhering" to the wall per unit time is given by the diffusion flux Ddw/Bx 
for x = 0, and the required probability W(t) over the time t is 



Substituting for to, we find 



W{t) = D^[dwldx] x = &t. 



W{t) = l-erf[* /2vW]- 



Problem 2. Determine the order of magnitude of the time t during which a particle 
suspended in a fluid turns through a large angle about its axis. 

Solution. The required time t is that during which a particle in Brownian motion moves 
over a distance of the order of its linear dimension a. According to (59.3) we have t r*s a 2 /D, 
and by (59.9) D ~ kT/ija. Thus t ~ 7]a s lkT. 



CHAPTER VII 

SURFACE PHENOMENA 

§60. Laplace's formula 

In this chapter we shall study the phenomena which occur near the surface 
separating two continuous media (in reality, of course, the media are separated 
by a narrow transitional layer, but this is so thin that it may be regarded as 
a surface). If the surface of separation is curved, the pressures near it in the 
two media are different. To determine the pressure difference (called the 
surface pressure), we write down the condition that the two media are in 
thermodynamic equilibrium together, taking into account the properties of 
the surface of separation. 

Let the surface of separation undergo an infinitesimal displacement. 
At each point of the undisplaced surface we draw the normal. The length of 
the segment of the normal lying between the points where it intersects the 
displaced and undisplaced surfaces is denoted by S£. Then a volume element 
between the two surfaces is S£d/, where d/ is a surface element. Let pi 
and/>2 be the pressures in the two media, and let S£ be reckoned positive if 
the displacement of the surface is towards medium 2 (say). Then the work 
necessary to bring about the above change in volume is 

j (-Pi+p2)8ldf. 

The total work 8R done in displacing the surface is obtained by adding to 
this the work connected with the change in area of the surface. This part of 
the work is proportional to the change S/in the area of the surface, and is aS/, 
where a is called the surface-tension coefficient.^ Thus the total work is 

8R = - j (p 1 -p 2 )8tdf+ a .8f (60.1) 

The condition of thermodynamic equilibrium is, of course, that 8R is zero. 
Next, let Ri and R% be the principal radii of curvature at a given point of 
the surface; we reckon R\ and R 2 as positive if they are drawn into medium 1. 
Then the elements of length d/i and d/ 2 on the surface in its principal sections 
receive increments (S£/i?i)d/i and (S£/i? 2 )d/ 2 respectively when the surface 
undergoes an infinitesimal displacement; here d/i and d/ 2 are regarded as 



f For an air-water interface a =72-5 erg/cm 2 at 20° C; for air and paraffin a = 24 at 20° C. 
The surface tension of liquid metals is very large; for instance, at an air-mercury interface a = 547 
at 175° C; for air and liquid platinum a = 1820 at 2000° C. The surface tension between liquid 
helium and its vapour is very small, a =0-24 at —270° C. 



230 



§60 Laplace's formula 231 

elements of the circumference of circles with radii JRi and R2. Hence the 
surface element d/ = d/id/2 becomes, after the displacement, 

d/i(l + 8£/12i)d&(l + 8£/122) « d/id&(l + 8£/22i + 8(7*2), 

i.e. it changes by S£d/(l/2?i+l/l?2)- Hence we see that the total change in 
area of the surface of separation is 

s/ =Mrt) d/ - (60 - 2) 

Substituting these expressions in (60.1) and equating to zero, we obtain 
the equilibrium condition in the form 



J«K^-(s + i)K-°- 



pi 



This condition must hold for every infinitesimal displacement of the surface, 
i.e. for all S£. Hence the expression in braces must be identically equal to 
zero: 

-»-'(k + k)- (60 - 3) 

This is Laplace's formula, which gives the surface pressure. We see that, 
if Ri and R2 are positive, pi — />2 > 0. This means that the pressure is greater 
in the medium whose surface is convex. If Ri = R 2 — 00, i.e. the surface 
of separation is plane, the pressure is the same in either medium, as we 
should expect. 

Let us apply formula (60.3) to investigate the mechanical equilibrium of 
two adjoining media. We assume that no external forces act, either on the 
surface of separation or on the media themselves. Using formula (60.3), we 
can then write the equation of equilibrium as 

1 1 

1 = constant. (60.4) 

Ri R2 

Thus the sum of the curvatures must be a constant over any free surface of 
separation. If the whole surface is free, the condition (60.4) means that it 
must be spherical (for instance, the surface of a small drop, for which the 
effect of gravity may be neglected). If, however, the surface is supported 
along some curve (for instance, a film of liquid on a solid frame), its shape is 
less simple. 

When the condition (60.4) is applied to the equilibrium of thin films 
supported on a solid frame, the constant on the right must be zero. For the 
sum 1/Ri+ l/i?2 must be the same everywhere on the free surface of the film, 
while on opposite sides of the film it must have opposite signs, since, if one 
side is convex, the other side is concave, and the radii of curvature are the 
same with opposite signs. Hence it follows that the equilibrium condition 



232 Surface Phenomena §60 

for a thin film is 

k + k=°- < 60 - 5 > 

Let us now consider the equilibrium condition on the surface of a medium 
in a gravitational field. We assume for simplicity that medium 2 is simply 
the atmosphere, whose pressure may be regarded as constant over the surface, 
and that medium 1 is an incompressible fluid. Then we havep2 = constant, 
while pi, the fluid pressure, is by (3.2) pi = constant - pgz, the co-ordinate 
z being measured vertically upwards. Thus the equilibrium condition 
becomes 

1 1 gpz 

— + — - H = constant. (60.6) 

Ri R% a 

It should be mentioned that, to determine the equilibrium form of the 
surface of the fluid in particular cases, it is usually convenient to use the 
condition of equilibrium, not in the form (60.6), but by directly solving the 
variational problem of minimising the total free energy. The internal free 
energy of an incompressible fluid depends only on the volume of the fluid, and 
not on the shape of its surface. The latter affects, firstly, the surface free 
energy J a d/ and, secondly, the energy in the external field (gravity), which 
* s gP J z dV. Thus the equilibrium condition can be written 

a J df+gp J zdV = minimum. (60.7) 

The minimum is to be determined subject to the condition 

f dV = constant, (60.8) 

which expresses the fact that the volume of the fluid is constant. 

The constants a, p and g appear in the equilibrium conditions (60.6) 
and (60.7) only in the form cc/gp. This ratio has the dimensions cm 2 . The 
length 

a = VV*lgp) (60.9) 

is called the capillary constant for the substance concerned.f The shape of 
the fluid surface is determined by this quantity alone. If the capillary 
constant is large compared with the dimension of the medium, we may 
neglect gravity in determining the shape of the surface. 

In order to find the shape of the surface from the condition (60.4) or 
(60.6), we need formulae which determine the radii of curvature, given the 
shape of the surface. These formulae are obtained in differential geometry, 



t For water (e.g.), a = 0-122 cm at 20° C. 



§60 



Laplace's formula 



233 



but in the general case they are somewhat complicated. They are consider- 
ably simplified when the surface deviates only slightly from a plane. We shall 
derive the appropriate formula directly, without using the general results of 
differential geometry. 

Let z = £(#, y) be the equation of the surface ; we suppose that £ is every- 
where small, i.e. that the surface deviates only slightly from the plane z = 0. 
As is well known, the area / of the surface is given by the integral 



or, for small £, approximately by 



The variation bf is 



'-JK(2H( 



8y 



dxdy. 



(60.10) 



)dx ay. 

8y dy ) 



8% 8% 



+ 



\8ldxdy. 



(60.11) 



8x 8x 
Integrating by parts, we find 

s '=-J"(2 

Comparing this with (60.2), we obtain 

1 1 _ / 8^ 8% 

#i + jfo~ ~\~8x~^~8f 

This is the required formula; it determines the sum of the curvatures of a 
slightly curved surface. 

When three adjoining media are in equilibrium, the surfaces of separation 
are such that the resultant of the surface-tension forces is zero on the common 
line of intersection. This condition implies that the surfaces of separation 
must intersect at angles (called angles of contact) determined by the values of 
the surface-tension coefficients.f 

Finally, let us consider the question of the boundary conditions that must 
be satisfied at the boundary between two fluids in motion, when the surface- 
tension forces are taken into account. If the latter forces are neglected, we 
have at the boundary between the fluids flj^a^.i* — vi,ik) = 0, which expresses 
the equality of the forces of viscous friction on the surface of each fluid. 
When the surface tension is included, we have to add on the right-hand 
side a force determined in magnitude by Laplace's formula and directed 
along the normal : 



Wfto^ift— nicotic 



( X l 
= a — + — 

\Ri R 2 



(60.12) 



t See, for instance, Statistical Physics, §145, Pergamon Press, London 1958. 



234 Surface Phenomena §60 

This equation can also be written 

(Pi-p2)tii = (a' 1>ik - ff' 2>tt )% + a — - + — )m. (60.13) 

If the two fluids are both ideal, the viscous stresses a' tk are zero, and we return 
to the simple equation (60.3). 

The condition (60.13), however, is still not completely general. The reason 
is that the surface-tension coefficient a may not be constant over the surface 
(for example, on account of a variation in temperature). Then, besides the 
normal force (which is zero for a plane surface), there is another force 
tangential to the surface. Just as there is a volume force -grad/> per unit 
volume (see §2) in cases where the pressure is not uniform, so we have here a 
tangential force f, = grad a per unit area of the surface of separation. In 
this case we take the positive gradient, because the surface-tension forces 
tend to reduce the area of the surface, whereas the pressure forces tend to 
increase the volume. Adding this force to the right-hand side of equation 
(60.13), we obtain the boundary condition 

T / l 1 \1 doc 

|/>i-Z>2-a^— + — J^m = (ff'i iir nttK+-; (60.14) 

the unit normal vector n is directed into medium 1. We notice that this 
condition can be satisfied only for a viscous fluid: in an ideal fluid, a' iJc = 
and the left-hand side of equation (60.14) is a vector along the normal, 
while the right-hand side is in this case a tangential vector. This equality 
cannot hold, except of course in the trivial case where both sides are zero. 



PROBLEMS 

Problem 1. Determine the shape of a film of liquid supported on two circular frames 
with their centres on a line perpendicular to their planes, which are parallel; Fig. 31 shows a 
cross-section of the film. 

Solution. The problem amounts to that of finding the surface having the smallest area 
that can be formed by the revolution about the line r = of a curve r — r(z) which passes 
between two given points A and B. The area of a surface of rotation is 

7m 

/dr\2- 



>->-NHl)Y 



§60 



Laplace's formula 



235 



It is well known that the minimum of an integral of the form 



J L(x, x) 



dt 



is given by the equation L — x dL/dx = constant. In the present case this leads to 

r = Cl V[l + (drldzf], 

whence we have by integration r = c x cosh[(«— c 2 )/cj. Thus the required surface (called a 
catenoid) is that formed by the revolution of a catenary. The constants c x and c 2 must be 
chosen so that the curve r{z) passes through the given points A and B. The value of c 2 
depends only on the choice of the origin of z. For the constant c lf however, two values are 
obtained, of which the larger must be chosen (the smaller does not give a minimum of the 
integral). 

When the distance h between the frames increases, it reaches a value for which the equation 
for the constant c x no longer has a real root. For greater values of h, only the shape consisting 
of one film on each frame is stable. For example, for two frames of equal radius R the catenoid 
form is impossible for a distance h between the frames greater than 1 -33.R. 

Problem 2. Determine the shape of the surface of a fluid in a gravitational field and 
bounded on one side by a vertical plane wall. The angle of contact between the fluid and the 
wall is 6 (Fig. 32). 



z 

l 






Fig. 32 



Solution. We take the co-ordinate axes as shown in Fig. 32. The plane x = is the plane 
of the wall, and z = is the plane of the fluid surface far from the wall. The radii of curvature 
of the surface z = z(x) are R x = oo, R 2 = -(1 +z' 2 ) i Jz", so that equation (60.6) becomes 



2z 



a* (1 + *' 2 )* 



= constant, 



(1) 



where a is the capillary constant. For * = oo we must have z = 0, 1/R 2 = 0, and the constant 
is therefore zero. A first integral of the resulting equation is 



1 



= A-— . 



(2) 



V(l + *' 2 ) a' 

From the condition at infinity (z = z' = for * = oo) we have A = 1. A second integration 
gives 

a . V2« 



+ 



•y(»-S) 



+ Xq. 



The constant * must be chosen so that, at the surface of the wall (x = 0), we have 
z' — —cot 9 or, by (2), z = h, where h = ay/{\— sin 0) is the height to which the fluid 
rises at the wall itself. 



236 



Surface Phenomena 



§60 



Problem 3. Determine the shape of the surface of a fluid rising between two parallel 
vertical flat plates (Fig. 33). 




Fig. 33 



Solution. We take the ys-plane half-way between the two plates, and the ay-plane to 
coincide with the fluid surface far from the plates. In equation (1) of Problem 2, which gives 
the condition of equilibrium and is therefore valid everywhere on the surface of the fluid 
(both between the plates and elsewhere), the conditions at x = oo again give the constant as 
zero. In the integral (2), the constant A is now different according as \x\ > id or Ixl < id 
(the function *(*) having a discontinuity for |x| = id). For the space between the plates, 
the conditions are z = for x = and *' = cot B for x = id, where $ is the angle of contact. 
According to (2) we have for the heights z = *(0) and x x = z(id): z = ax/(A-l) 
Zi = aV(A—sm 0). Integrating (2), we obtain 






So 



(A-z 2 /a*)dz 
V[l-(-4-# 2 /« 2 ) 2 ] 



aV(A-cos gy=z 



= \a 



\ 



cos i dg 



V(A-cos£y 



where ^is a new variable related to z by z = a y/(A -cos i). This is an elliptic integral, and 
cannot be expressed in terms of elementary functions. The constant A is found from the 
condition that z — z t for x = \d, or 



in-6 



..; 



cos f d£ 
V(^-cos£)' 



The formulae obtained above give the shape of the fluid surface in the space between the 
plates. As d -*• 0, A tends to infinity. Hence we have for d <^ a 



VA 



tn—u 

cos£d£ = — — cos0, 

J \/A 



or A = (a 2 /d 2 ) cos 2 0. The height to which the fluid rises is z X z t X (a 2 /d) cos 9; this 
formula can also be obtained directly, of course. 

Problem 4. A thin non-uniformly heated layer of fluid rests on a horizontal plane solid 
surface; its temperature is a given function of the co-ordinate x in the plane, and (because 
the layer is thin) may be supposed independent of the co-ordinate z across the layer. The 
non-uniform heating results in the occurrence of a steady flow, and its thickness £ con- 
sequently varies in the x-direction. Determine the function C(x). 

Solution. The fluid density p and the surface tension a are, together with the temperature 
known functions of x. The fluid pressure p = Po + pgtf-z), where p is the atmospheric 



§61 Capillary waves 237 

pressure (the pressure on the free surface) ; the variation of pressure due to the curvature of the 
surface may be neglected. The fluid velocity in the layer may be supposed everywhere parallel 
to the »-axis. The equation of motion is 

rjd^/dz 2 = dpjdx = g[d(p£)ldx-zd P ldx]. (1) 

On the solid surface (z = 0) we have v = 0, while on the free surface (z = £) the boundary 
condition (60.14) must be fulfilled; in this case it is rj[dv/dz] z <=z = da/dx. Integrating equa- 
tion (1) with these conditions, we obtain 

ijv = gz(i;-lz)d(pi;)ldx-igz(3P-zZ)dpldx-zdoLldx. (2) 

Since the flow is steady, the total mass flux through a cross-section of the layer must be 
zero : 



jvdz = 0. 



Substituting (2), we find 



dt 2 dp 1 da 

In + ££2_L t 

dx dx g dx 



3A 
which determines the function £(x). Integrating, we obtain 

gt 2 = 3/>-*[f/r-*da + constant]. (3) 

If the temperature (and therefore p and a) varies only slightly, then (3) can be written 

P = Wpolp)* + %*-«o)lf>g, 
where £ is the value of £ at a point where p = p ^d a = a o- 

§61. Capillary waves 

Fluid surfaces tend to assume an equilibrium shape, both under the action 
of the force of gravity and under that of surface-tension forces. In studying 
waves on the surface of a fluid in §§12 and 13, we did not take the latter forces 
into account. We shall see below that capillarity has an important effect on 
gravity waves of small wavelength. 

As in §12, we suppose the amplitude of the oscillations small compared 
with the wavelength. For the velocity potential we have as before the equa- 
tion A^ = 0. The condition at the surface of the fluid is now different, 
however: the pressure difference between the two sides of the surface is 
not zero, as we supposed in §12, but is given by Laplace's formula (60.3). 

We denote by I the z co-ordinate of a point on the surface. Since £ is 
small, we can use the expression (60.11), and write Laplace's formula as 

\ dx 2 By 2 

Here p is the pressure in the fluid near the surface, and po is the constant 
external pressure. For p we substitute, according to (12.2), 

P = -pgl-pm^t, 



238 Surface Phenomena §61 

obtaining 

Pgt+p— - a + — =0; 

dt \dx* dyZJ 

for the same reasons as in §12, we can omit the constant p if we redefine <f>. 
Differentiating this relation with respect to t, and replacing dt,\dt by tyjdz, 
we obtain the boundary condition on the potential <f>: 

d<f> 8U d / 8U d 2 4>\ 

'^ + ^-"&(^ + vH for * =0 - (6U) 

Let us consider a plane wave propagated in the direction of the #-axis. 
As in §12, we obtain a solution in the form <f> = Ae kz cos(kx-cot). The 
relation between k and o> is now obtained from the boundary condition 
(61.1), and is 

co 2 =gk + xkZjp. (61.2) 

We see that, for long wavelengths such that k ^ V{gpl<*), or k ^ \\a 

(where a is the capillary constant), the effect of capillarity may be neglected, 

and we have a pure gravity wave. In the opposite case of short wavelengths, 

the effect of gravity may be neglected. Then 

o>2 = ajfi/p, ( 61 3 ) 

Such waves are called capillary waves or ripples. Intermediate cases are 
referred to as capillary gravity waves. 

Let us also determine the nature of the oscillations of a spherical drop of 
incompressible fluid under the action of capillary forces. The oscillations 
cause the surface of the drop to deviate from the spherical form. As usual, we 
shall suppose the amplitude of the oscillations to be small. 

We begin by determining the value of the sum l/i?i+ 1/R 2 for a surface 
slightly different from that of a sphere. Here we proceed as in the derivation 
of formula (60.11). The area of a surface given in spherical co-ordinatesf 
r, 6, <f> by a function r = r{6, <f>) is 



T 

A spherical surface is given by r = constant = R (where R is the radius 
of the sphere), and a neighbouring surface by r = R+£, where £ is small. 
Substituting in (61.4), we obtain 



'■'m^m^Q' 



sin0d0d^. 



t In the remainder of this section </> denotes the azimuthal angle, and we denote the velocity 
potential by tfi. 



§61 Capillary waves 239 

Let us find the variation 6/ in the area when £ changes. We have 

2 f H dl dhl 1 dl Bht, ) 

bf = \{2(R + mt + — — - + -— - sin0d0d<£. 

J J J I K } 36 dd sin20 d<f> d<f> I Y 



Integrating the second term by parts with respect to 0, and the third by 
parts with respect to </>, we obtain 

rVr 13/ an 1 a 2 n 

8f= 2CR + £) s in0— -^— -J3£sin0d0d<£ 

J J J\ K ' sin0 dd\ 86 sin20 ^2 J 





If we divide the expression in braces by R(R + 2Q, the resulting coefficient 
of S£S/ « 8£R(R+ 2£) sin dddd<f> in the integrand is, by formula (60.2), 
just the required sum of the curvatures, correct to terms of the first order in £. 
Thus we find 

± + ± = !_?£_l(_i-f£ + _L!(si„^)l. (6i.5) 

R x R 2 R R 2 # 2 lsin20 d<p sin0 dd \ ddl) 

The first term corresponds to a spherical surface, for which JRi = R% = R. 
The velocity potential if/ satisfies Laplace's equation A^ = 0, with a 
boundary condition atr = R like that for a plane surface: 

dip (2 21 l r l a / an l a^-n 

pJL + J sin0— + -— ■ — \\+po = 0. 

P a* 1/2 /?2 /?2Lsin0a0\ 50/ sin20 a^2jj 

The constant po + 2«.jR can again be omitted; differentiating with respect to 
time and putting dlfdt = v r = a«/»/ar, we have finally the boundary condition 
on^: 



P 



dp RH dr arLsin0a0\ 30 J sin 2 d<f>*]\ 



for r = R. (61.6) 

We shall seek a solution in the form of a stationary wave: tp = er+tflr, 0, ^), 
where the function /satisfies Laplace's equation, A/ = 0. As is well known, 
any solution of Laplace's equation can be represented as a linear combination 
of what are called volume spherical harmonic functions r l Y lm (9, <f>), where 
Y lm (d,<f>) are Laplace's spherical harmonics: Y lm (d, <f>) = P, w (cos 6)e im< t>. 
Here P, m (cos 0) = sin m d m Pi (cos 0)/d (cos 0) m is what is called an associated 
Legendre function, Pi (cos 0) being the Legendre polynomial of order /. As 
is well known, / takes all integral values from zero upwards, while m takes the 
values 0, ±1, ±2, ..., ±1 

Accordingly, we seek a particular solution of the problem in the form 

tf, = Ae- i(0t r l Pi m (cos dy™*. (61.7) 



240 Surface Phenomena §61 

The frequency co must be such as to satisfy the boundary condition (61.6). 
Substituting the expression (61.7) and using the fact that the spherical har- 
monics Y lm satisfy 

1 3 / . J Y im\ 1 8*Yi m 



sin0 dd\ 86/ sin 2 302 

we find (cancelling ift) 

pa> + fo[2-l(l+l)]IR? = 0, 



or 



o? = o/(/-l)(/+2)/p/28. (61.8) 

This formula gives the eigenfrequencies of capillary oscillations of a 
spherical drop. We see that it depends only on /, and not on m. To a given /, 
however, there correspond 21+ 1 different functions (61.7). Thus each 
of the frequencies (61.8) corresponds to 21+ 1 different oscillations. Inde- 
pendent oscillations having the same frequency are said to be degenerate; 
in this case we have (21+ l)-fold degeneracy. 

The expression (61.8) vanishes for / = and / = 1. The value / = 
would correspond to radial oscillations, i.e. to spherically symmetrical pulsa- 
tions of the drop; in an incompressible fluid such oscillations are clearly 
impossible. For / = 1 the motion is simply a translatory motion of the drop 
as a whole. The smallest possible frequency of oscillations of the drop cor- 
responds to / = 2, and is 

w m in = V(8a//># 3 ). (61.9) 

A peculiar wave motion due to surface tension is observed when a thin 
layer of viscous fluid flows down a vertical wall. P. L. Kapitza has shown that 
these waves must be due to an instability of the original flow that sets in at 
comparatively small Reynolds numbers.f 

PROBLEMS 

Problem 1. Determine the frequency as a function of the wave number for capillary 
gravity waves on the surface of a fluid of depth h. 

Solution. Substituting in the condition (61.1) <j> = A cos(kx-o)t) cosh k(z+h) (cf. §12, 
Problem 1), we obtain co 2 = (gk+a.k 3 Jp) tanh kh. For tt>lwe return to formula (61.2)', 
while for long waves (kh <^ 1) we have to 2 = ghk 2 + a.hk i Jp. 

Problem 2. Determine the damping coefficient for capillary waves. 

Solution. Substituting (61.3) in (25.5), we find y = 2 v k 2 /p = 2i?w 4 / 3 /p 1/3 a 2/s . 

Problem 3. Find the condition for the stability of a horizontal tangential discontinuity in a 
gravitational field, taking account of surface tension (the fluids on the two sides of the sur- 
face of discontinuity being supposed different). 

Solution. Let U be the velocity of the upper fluid relative to the lower. On the original 
flow we superpose a perturbation periodic in the horizontal direction, and seek the velocity 



t See P. L. Kapitza, Zhurnal eksperimental'nol i teoreticheskoi fiziki 18, 3, 1948. 



§62 The effect of adsorbed films on the motion of a liquid 241 

potential in the form 

<f> = Ae kz COs(&tf — 0)t) in the lower fluid, 

<£' = A'e~ kz COS(&V— COt)+ Ux in the upper fluid. 

For the lower fluid we have on the surface of discontinuity v z = d<j>Jdz = d£/d£, where t, is a 
vertical co-ordinate in the surface of discontinuity, and for the upper fluid 

v ' z = dtffdz = Udydx+dt/dt. 

The condition of equal pressures in the two fluids at the surface of discontinuity is 

P d4>ldt+pgl-*Ptld& = P 'd<f>'ldt+ P 'gt:+ip\v' 2 -U 2 ); 

only terms of the first order in A' need be retained in expanding the expression v'* — U 2 . 
We seek the displacement £ in the form t, = asin(kx — cat). Substituting <f>, <f>' and £ in 
the above three conditions for z — 0, we obtain three equations from which a, A and A' 
can be eliminated, leaving 

tyU l\kg{p- P ') tfpp'UZ a& 

CO = 



l + I\ kg(p-P) Wpp'U* | a#» I 

/ ± VL p+p {p+pf p+p y 



p+p 

In order that this expression should be real for all k, it is necessary that 

V A ^ 4* g (p-p')(p+p'ripy 2 . 

If this condition does not hold, there are complex a> with a positive imaginary part, and the 
motion is unstable. 

§62. The effect of adsorbed films on the motion of a liquid 

The presence on the surface of a liquid of a film of adsorbed material may 
have a considerable effect on the hydrodynamical properties of the surface. 
The reason is that, when the shape of the surface changes with the motion of 
the liquid, the film is stretched or compressed, i.e. the surface concentration 
of the adsorbed substance is changed. These changes result in the appearance 
of additional forces which have to be taken into account in the boundary 
conditions at the free surface. 

Here we shall consider only adsorbed films of substances which may be 
regarded as insoluble in the liquid. This means that the substance is entirely 
on the surface, and does not penetrate into the liquid. If the adsorbed 
substance is appreciably soluble, it is necessary to take into account the 
it diffusion of between the surface film and the volume of the liquid when the 
concentration of the film varies. 

When the adsorbed material is present, the surface-tension coefficient a 
is a function of the surface concentration of the material (the amount of it 
per unit surface area), which we denote by y. If y varies over the surface, 
then the coefficient a is also a function of the co-ordinates in the surface. 
The boundary condition at the surface of the liquid therefore includes 
a tangential force, which we have already discussed at the end of §60 (equation 
(60.14)). In the present case, the gradient of a can be expressed in terms of 
the surface concentration gradient, so that the tangential force on the surface is 

ft = (3a/dy)grady. (62.1) 



242 Surface Phenomena §62 

It has been mentioned in §60 that the boundary condition (60.14), in which 
this force is taken into account, can be satisfied only for a viscous fluid. 
Hence it follows that, in cases where the viscosity of the liquid is small, and 
unimportant as regards the phenomenon under consideration, the presence 
of the film can be ignored. 

To determine the motion of a liquid covered by a film we must add to the 
equations of motion, with the boundary condition (60.14), a further equation, 
since we now have another unknown quantity, the surface concentration y. 
This further equation is an "equation of continuity", expressing the fact that 
the total amount of adsorbed material in the film is unchanged. The actual 
form of the equation depends on the shape of the surface. If the latter is 
plane, then the equation is evidently 

dyldt + 8(yv x )/8x + 8(yv y )/8y = 0, (62.2) 

where all quantities have their values at the surface (taken as the ay-plane). 

The solution of problems of the motion of a liquid covered by an adsorbed 
film is considerably simplified in cases where the film may be supposed 
incompressible, i.e. we may assume that the area of any surface element of the 
film remains constant during the motion. 

An example of the important hydrodynamic effects of an adsorbed film is 
given by the motion of a gas bubble in a viscous liquid. If there is no film 
on the surface of the bubble, the gas inside it moves also, and the drag force 
exerted on the bubble by the liquid is not the same as the drag on a solid 
sphere of the same radius (see §20, Problem 2). If, however, the bubble is 
covered by a film of adsorbed material, it is clear from symmetry that the 
film remains at rest when the bubble moves. For a motion in the film could 
occur only along meridian lines on the bubble surface, and the result would be 
that material would continually accumulate at one of the poles (since the 
adsorbed material does not penetrate into the liquid or the gas); this is 
impossible. Besides the velocity of the film, the gas velocity at the surface 
of the bubble must also be zero, and with this boundary condition the gas 
in the bubble must be entirely at rest. Thus a bubble covered by a film moves 
like a solid sphere and, in particular, the drag on it (for small Reynolds 
numbers) is given by Stokes' formula. This result is due to V. G. Levich, 
who also gave the solutions to the following Problems.f 

PROBLEMS 

Problem 1. Two vessels are joined by a long deep channel of width a and length / with 
plane parallel walls. The surface of the liquid in the system is covered by an adsorbed film, 
and the surface concentrations y x and y 2 of the film in the two vessels are different. There 
results a motion near the surface of the liquid in the channel. Determine the amount of 
film material transported by this motion. 

Solution. We take the plane of one wall of the channel as the xz-plane, and the surface 
of the liquid as the #y-plane, so that the *-axis is along the channel; the liquid is in the region 



t For a more detailed account see V. G. Levich, Physico-chemical Hydrodynamics (Fiziko- 
khimicheskaya gidrodinamika), Moscow 1952. 



§62 The effect of adsorbed films on the motion of a liquid 243 

z < 0. There is no pressure gradient, so that the equation of steady flow is (cf. §17) 

d 2 v d 2 v 

+ = o, (1) 

dy* 3*2 

where v is the liquid velocity, which is evidently in the ac-direction. There is a concentration 
gradient dy/d* along the channel. At the surface of the liquid in the channel we have the 
boundary condition 

7) dvjdz = doc/dx for z = 0. (2) 

At the channel walls the liquid must be at rest, i.e. 

V = for y = and y = a. (3) 

The channel depth is supposed infinite, and so 

V = for Z -> — 00. (4) 

Particular solutions of equation (1) which satisfy the conditions (3) and (4) are 
v n — constant X exp[(2« + l)iTz[a] sin(2»+l)7ry/a, 

with n integral. The condition (2) is satisfied by the sum 

4a da^ exp[(2«+l)7r#/a] sm{2n + \)iryja 
yS^ (2« + l)2 ' 

The amount of film material transferred per unit time is 

f 8a 2 / ^ 1 \ da 

the motion being in the direction of a increasing. The value of Q must obviously be constant 
along the channel. Hence we can write 

da If da l? 1 , 

y — = constant = - y— dx= - yd<x, 
dx I J dx I J 

«s 

where a x = a(y x ), <% == a(y 2 ), and we assume that a x > a^ Thus we have finally 

8a 2 /^ 1 \ f a 2 f 1 

Q = ( > y da = 0-27— y da. 

T ? /7r3\^(2«+l)3/ J Y r)lj r 



Problem 2. Determine the damping coefficient for capillary waves on the surface of a 
liquid covered by an adsorbed film. 

Solution. If the viscosity of the liquid is not too great, the stretching (tangential) forces 
exerted on the film by the liquid are small, and the film may therefore be regarded as in- 
compressible. Accordingly, we can calculate the energy dissipation as if it took place at a 



244 Surface Phenomena §62 

solid wall, i.e. from formula (24.14). Writing the velocity potential in the form 

eh = J)q gikx—iat g—kz 
we obtain for the dissipation per unit area of the surface 

T 

En.m = - ^{lp-t]0))\hj>o\ 2 . 
The total energy (also per unit area) is 

e = p\^dz = y\k^ik. 

The damping coefficient is (using (61.3)) 

0,7/<y/2 £7/4^1/2^/4 

2^2<x.v s p 1/6 ~ 2V2p 3 ' 4 ' 

The ratio of this quantity to the damping coefficient for capillary waves on a clean surface 
(§61, Problem 2) is (a/>/£i? 2 ) l/4 /4V2, and is large compared with unity unless the wavelength 
is extremely small. Thus the presence of an adsorbed film on the surface of a liquid leads to 
a marked increase in the damping coefficient. 



7 = 



CHAPTER VIII 

SOUND 

§63. Sound waves 

We proceed now to the study of the flow of compressible fluids, and begin 
by investigating small oscillations; an oscillatory motion of small amplitude 
in a compressible fluid is called a sound wave. At each point of the fluid, 
a sound wave causes alternate compression and rarefaction. 

Since the oscillations are small, the velocity v is small also, so that the term 
(v«grad)v in Euler's equation may be neglected. For the same reason, the 
relative changes in the fluid density and pressure are small. We can write 
the variables p and p in the form 

p=po+p'> p = po+p, (63.1) 

where po and po are the constant equilibrium density and pressure, and p 
and/>' are their variations in the sound wave (p <^ po, p' -4 Po). The equation 
of continuity dpldt+div(pv) = 0, on substituting (63.1) and neglecting small 
quantities of the second order (/>', p' and v being of the first order), becomes 

dp'/dt+podivv = 0. (63.2) 

Euler's equation 

3v/3* + (vgrad)v = -(l/p)gradp 
reduces, in the same approximation, to 

0v/0*+(l/po)gradp' = 0. (63.3) 

The condition that the linearised equations of motion (63.2) and (63.3) 
should be applicable to the propagation of sound waves is that the velocity 
of the fluid particles in the wave should be small compared with the velocity 
of sound: v <^ c. This condition can be obtained, for example, from the 
requirement that p' <^ po (see formula (63.12) below). 

Equations (63.2) and (63.3) contain the unknown functions v, p' and p' . 
To eliminate one of these, we notice that a sound wave in an ideal fluid is, 
like any other motion in an ideal fluid, adiabatic. Hence the small change 
p' in the pressure is related to the small change p' in the density by 

p' = (dp!d P o) s p f . (63.4) 

Replacing p according to this equation in (63.2), we find 

dp'l8t+po(dpldp ) s divv = 0. (63.5) 

The two equations (63.3) and (63.5), with the unknowns v and p', give a 
complete description of the sound wave. 

9 245 



246 Sound §63 

In order to express all the unknowns in terms of one of them, it is con- 
venient to introduce the velocity potential by putting v = grad <f>. We 
have from equation (63.3) 

p' = -ptyjdt, (63.6) 

which relates p' and (f> (here, and henceforward, we omit for brevity the 
suffix inpo and po). We then obtain from (63.5) the equation 

dmdt2-c*A<f> = 0, (63.7) 

which the potential <f> must satisfy; here we have introduced the notation 

c = Vilify)*. (63.8) 

An equation of the form (63.7) is called a wave equation. Applying the 
gradient operator to (63.7), we find that each of the three components of the 
velocity v satisfies an equation of the same form, and on differentiating 
(63.7) with respect to time we see that the pressure p' (and therefore p) 
also satisfies the wave equation. 

Let us consider a sound wave in which all quantities depend on only one 
co-ordinate (x, say). That is, the flow is completely homogeneous in the 
ys-plane. Such a wave is called a. plane wave. The wave equation (63.7) 
becomes 

8P4>/dx*-(llc*)dmdt* = 0. (63.9) 

To solve this equation, we replace x and t by the new variables £ = x — ct, 
t] = x+ct. It is easy to see that in these variables (63.9) becomes 
d 2 (f>ldir]dg = 0. Integrating this equation with respect to £, we find 
8<f>jdr) = F(rf), where F(rj) is an arbitrary function of 17. Integrating again, 
we obtain <f> = /i(£) +/2(i?)i where /1 and fa are arbitrary functions of their 
arguments. Thus 

<f> =fi(x-ct)+f 2 (x+ct). (63.10) 

The distribution of the other quantities (p', />', v) in a plane wave is given by 
functions of the same form. 

For definiteness, we shall discuss the density, p = fi(x—ct)+f2(x+ct). 
For example, let/2 = 0, so that p = fi(x—cf). The meaning of this solution 
is evident: in any plane x = constant the density varies with time, and at any 
given time it is different for different x, but it is the same for co-ordinates x 
and times t such that x—ct= constant, or x = constant + ct. This means 
that, if at some instant t — and at some point the fluid density has a certain 
value, then after a time t the same value of the density is found at a distance 
ct along the #-axis from the original point. The same is true of all the other 
quantities in the wave. Thus the pattern of motion is propagated through 
the medium in the ^-direction with a velocity c; c is called the velocity of 
sound. 

Thus/i(«— ct) represents what is called a travelling plane wave propagated 
in the positive direction of the #-axis. It is evident that/^+c*) represents 
a wave propagated in the opposite direction. 



§63 Sound waves 247 

Of the three components of the velocity v = grad in a plane wave, only 
v x — d^Jdx is not zero. Thus the fluid velocity in a sound wave is in the 
direction of propagation. For this reason sound waves in a fluid are said to 
be longitudinal. 

In a travelling plane wave, the velocity v x — v is related to the pressure p' 
and the density p in a simple manner. Putting <f> = f(x—ct), we find 
v — d<f>Jdx = f'(x—ci) and p' — — pd<j>jdt = pcf'{x—ct). Comparing the 
two expressions, we find 

v = p'/pc. (63.11) 

Substituting here from (63.4) p' = c 2 p, we find the relation between the 
velocity and the density variation : 

v = cp'/p. (63.12) 

We may mention also the relation between the velocity and the temperature 
oscillations in a sound wave. We have 7" = {dT\dp\p' and, using the well- 
known thermodynamic formula (dT[dp) s = (Tlcp)(dV/dT)p and formula 
(63.11), we obtain 

T = tfTv/cp, (63.13) 

where /? = {\jV){dVjdT) p is the coefficient of thermal expansion. 

Formula (63.8) gives the velocity of sound in terms of the adiabatic 
compressibility of the fluid. This is related to the isothermal compressibility 
by the thermodynamic formula 

(dp/dp), = (c P lc v )(8pldp) T . (63.14) 

Let us calculate the velocity of sound in a perfect gas. The equation of state 
ispV = pip = RT/p., where R is the gas constant and p. the molecular weight. 
We obtain for the velocity of sound the expression 

c = V(yRT/p,), (63.15) 

where y denotes the ratio Cp\c v .\ Since y usually depends only slightly on the 
temperature, the velocity of sound in the gas may be supposed proportional 
to the square root of the temperature. For a given temperature it does not 
depend on the pressure. 

What are called monochromatic waves are a very important case. Here all 
quantities are just periodic (harmonic) functions of the time. It is usually 
convenient to write such functions as the real part of a complex quantity (see 
the beginning of §24). For example, we put for the velocity potential 

<f> = re[<f>o(x,y,z)e-^l (63.16) 

where w is the frequency of the wave. The function <£o satisfies the equation 

Ah + Mc^o = 0, (63.17) 

which is obtained by substituting (63.16) in (63.7). 



t It is useful to note that the velocity of sound in a gas is of the same order of magnitude as the 
mean thermal velocity of the molecules. 



248 Sound §63 

Let us consider a monochromatic travelling plane wave, propagated in the 
positive direction of the #-axis. In such a wave, all quantities are functions 
oi x—ct only, and so the potential is of the form 

cf> = re{^4 exp[-ico(t-x/c)]} y (63.18) 

where A is a constant called the complex amplitude. Writing this as A = ae i<x 
with real constants a and a, we have 

(f> = acos(coxlc—cot + <x). (63.19) 

The constant a is called the amplitude of the wave, and the argument of the 
cosine is called the phase. We denote by n a unit vector in the direction of 
propagation. The vector 

k = (o)/c)n = (27r/A)n (63.20) 

is called the wave vector. In terms of this vector (63.18) can be written 

<j> = re{A exp[)'(k.r- cot)]}. (63.21) 

Monochromatic waves are very important, because any wave whatsoever 
can be represented as a sum of superposed monochromatic plane waves 
with various wave vectors and frequencies. This decomposition of a wave 
into monochromatic waves is simply an expansion as a Fourier series or inte- 
gral (called also spectral resolution). The terms of this expansion are called 
the monochromatic components or Fourier components of the wave. 

PROBLEMS 

Problem 1 . Determine the velocity of sound in a nearly homogeneous two-phase system 
consisting of a vapour with small liquid droplets suspended in it (a "wet vapour"), or a liquid 
with small vapour bubbles in it. The wavelength of the sound is supposed large compared 
with the size of the inhomogeneities in the system. 

Solution. In a two-phase system, p and T are not independent variables, but are related 
by the equation of equilibrium of the phases. A compression or rarefaction of the system is 
accompanied by a change from one phase to the other. Let x be the fraction (by mass) of 
phase 2 in the system. We have 

S = (l-x)Si + XS2y 

V = (l-x)Vi + xV 2 , ( 

where the suffixes 1 and 2 distinguish quantities pertaining to the pure phases 1 and 2. To 
calculate the derivative (dV/dp)„ we transform it from the variables p, s to p, x, obtaining 
(8V/8p), = (dV/8p)x—(dVjdx) ]> (dsldp)xK8sldx) ]> . The substitution (1) then gives 

l!L\ = J^_^zZl^1 +(1 _4^l_-^zZl^1. (2) 

\ dp I s L dp s 2 -si dp J L dp s 2 -si dp J 

The velocity of sound is obtained from (1) and (2), using formula (63.8). 

Expanding the total derivatives with respect to the pressure, introducing the latent heat 
of the transition from phase 1 to phase 2 (q = T(s 2 — s x )), and using the Clapeyron-Clausius 
equation for the derivative dp/dT along the curve of equilibrium (dp/dT = p/T(V 2 — Vi)), 
we obtain the expression in the first brackets in (2) in the form 

/ 8V 2 \ 2T I dV 2 \ Tcj? 

(-wi + r(^i {V2 - Vl) -^- Vl)K 

The second bracket is transformed similarly. 



§64 The energy and momentum of sound waves 249 

Let phase 1 be the liquid and phase 2 the vapour; we suppose the latter to be a perfect 
gas, and neglect the specific volume V t in comparison with V z . If * <^ 1 (a liquid containing 
some bubbles of vapour), the velocity of sound is found to be 

c = qppVxIRTy/ifaT), (3) 

where jR is the gas constant and p the molecular weight. This velocity is in general very 
small ; thus, when vapour bubbles form in a liquid {cavitation), the velocity of sound undergoes 
a sudden sharp decrease. 

If 1 — x <^ 1 (a vapour containing some droplets of liquid), we obtain 

1 ix, 2 C V ,T 

c 2 RT q q* v ' 

Comparing this with the velocity of sound in the pure gas (63.15), we find that here also the 
addition of a second phase reduces the value of c, though by no means so markedly. 

As * increases from to 1 , the velocity of sound increases monotonically from the value (3) 
to the value (4). For # = and * = 1 it changes discontinuously as we go from a one-phase 
system to a two-phase system. This has the result that, for values of x very close to zero or 
unity, the usual linear theory of sound is no longer applicable, even when the amplitude of 
the sound wave is small; the compressions and rarefactions produced by the wave are in 
this case accompanied by a change between a one-phase and a two-phase system, and the 
essential assumption of a constant velocity of sound no longer holds good. 

Problem 2. Determine the velocity of sound in a gas heated to such a high temperature 
that the pressure of equilibrium black-body radiation becomes comparable with the gas 
pressure. 

Solution. The pressure is p = nkT+iakT 4 , and the entropy is 

s = (klm)\og(T*/n) + akT3ln. 

In these expressions the first terms relate to the particles, and the second terms to the radia- 
tion; n is the number density of particles, m their mass, k Boltzmann's constant, and 
a = 4ir 2 k?l45h 3 c 3 .\ The density of matter is not affected by the black-body radiation, so that 
P = mn. The velocity of sound, denoted here by w to distinguish it from that of light, is 

d(p,s) d(p yS ) i d( P ,s) 

u 2 = 



/■ 



d(p,s) d(n,T)l d(n,T) 

where the derivatives have been written in Jacobian form. Evaluating the Jacobians, we have 

SkT r 2*276 -, 

m 2 = 1+ . 

3m L 5n(n + 2aT3)\ 

§64. The energy and momentum of sound waves 

Let us derive an expression for the energy of a sound wave. According to 
the general formula, the energy in unit volume of the fluid is pe + ^pv 2 . 
We now substitute p = po + p', e = eo + e', where the primed letters denote 
the deviations of the respective quantities from their values when the fluid 
is at rest. The term %pv 2 is a quantity of the third order. Hence, if we take 
only terms up to the second order, we have 

poco+p — — + ip 2 — — - + ipov 2 . 

opo Opo* 



f See, for instance, Statistical Physics, §60, Pergamon Press, London 1958. 



250 Sound §64 

The derivatives are taken at constant entropy, since the sound wave is adiaba- 
tic. From the thermodynamic relation de = Tds—pdV = Tds+(p}p 2 )dp we 
have [d(p€)fdp] s = c+pjp = w, and the second derivative is 

[82(pe)l8p] s = (8w/8p) s = (dw/dp) s (8pldp) s = c*\p. 

Thus the energy in unit volume of the fluid is 

/>o*o + wop' + hc 2 p 2 /p + Ipov 2 . 

The first term (/»oeo) in this expression is the energy in unit volume when 
the fluid is at rest, and does not relate to the sound wave. The second term 
(toop) is the change in energy due to the change in the mass of fluid in unit 
volume. This term disappears in the total energy, which is obtained by 
integrating the energy over the whole volume of the fluid: since the total mass 
of fluid is unchanged, we have 

jpdV=j Po dV, or j p' dV = 0. 

Thus the total change in the energy of the fluid caused by the sound wave is 
given by the integral 

f (ip0V* + fr2p'2/p )dV. 

The integrand may be regarded as the density E of sound energy: 

E = Jpooa+fcya/po. (64.1) 

This expression takes a simpler form for a travelling plane wave. In such 
a wave p = po v\c (see (63.12)), and the two terms in (64.1) are equal, so that 

E = p v z . (64.2) 

In general this relation does not hold. A similar formula can be obtained only 
for the (time) average of the total sound energy. It follows immediately from 
a well-known general theorem of mechanics, that the mean total potential 
energy of a system executing small oscillations is equal to the mean total 
kinetic energy. Since the latter is, in the case considered, 

ijpo^dr, 

we find that the mean total sound energy is 

j EdV = j potfdV. (64.3) 

If a non-monochromatic wave is represented as a series of monochromatic 
waves, the mean energy is equal to the sum of the mean energies of the 
monochromatic components. For, if v is represented as a sum of terms of 



§64 The energy and momentum of sound waves 25 1 

various frequencies, v 2 will contain both the square of each term and the 
products of terms of different frequencies. These products contain factors of 
the form «*<"-*»'>', which are periodic functions of time. But the mean value 
of a periodic function is zero, and these terms therefore vanish. Thus the 
mean energy contains only terms in the mean squares of the monochromatic 
components. 

Next, let us consider some volume of a fluid in which sound is propagated, 
and determine the mean flux of energy through the closed surface bounding 
this volume. The energy flux density in the fluid is, by (6.3), pv(£v 2 +w). 
In the present case we can neglect the term in v 2 y which is of the third order. 
Hence the mean energy flux density in the sound wave is pzov. Substituting 
w = wo + w', we have pwv = zoopv+pw'v. For a small change w' in the 
heat function we have w' = (dzv/dp) s p'. Since {dwjdp) 8 = 1/p, it follows that 
to' = p'jp and pzov = zoopv+p'v. The total energy flux through the surface 
in question is 

<j> (wopv+'pv)'df. 

However, since the total quantity of fluid in the volume considered is un- 
changed on the average, the time average of the mass flux through the closed 
surface must be zero. Hence the energy flux is simply 



(U'vdf. 



We see that the mean sound energy flux is represented by the vector 

q = ~pv. (64.4) 

It is easy to verify that the relation 

dE/dt + div(p'v) = (64.5) 

holds. In this form the equation gives the law of conservation of the sound 
energy, with the vector q = p'v taking the part of the sound energy flux. 
Thus the expression is valid not only for the mean flux but also for the flux 
at any instant. 

In a travelling plane wave the pressure variation is related to the velocity 
by P' — c PoV' Introducing the unit vector n in the direction of propagation 
of the wave (which is the same as the direction of the velocity v), we obtain 
q = cpov 2 n, or 

q = cEn. (64.6) 

Thus the energy flux density in a plane sound wave equals the energy density 

multiplied by the velocity of sound, a result which was to be expected. 

Let us now consider a sound wave which, at any given instant, occupies a 

finite region of spacef (a wave packet), and determine the total momentum of 



f Nowhere bounded by solid walls. 



252 Sound §64 

the fluid in the wave. The momentum of unit volume of fluid is equal to the 
mass flux density j = p\. Substituting p = po + p', we have j = pov+p'v. 
The density change is related to the pressure change by p = p'jc 2 . Using 
(64.4), we therefore obtain 

j = pov+q/c*. (64.7) 

Since we have potential flow in a sound wave, we can write v = grad <£ ; it 
should be emphasised that this result is not a consequence of the approxi- 
mations made in deriving the linear equations of motion in §63, since a solu- 
tion such that curl v = is an exact solution of Euler's equations. We 
therefore have j = po grad + q/c 2 . The total momentum in the wave equals 
the integral J j dV over the volume occupied by the wave. The integral of 
grad <f> can be transformed into a surface integral, 



j grad<f>dV = j><f>df, 



and is zero, since <f> is zero outside the volume occupied by the wave. Thus the 
total momentum of the wave is 

JjdF = (1/^2) j qdF. (64.8) 

This quantity is not, in general, zero. The existence of a non-zero total 
momentum means that there is a transfer of matter. We therefore conclude 
that the propagation of a sound-wave packet is accompanied by the transfer 
of fluid. This is a second-order effect (since q is a second-order quantity). 

Finally, let us calculate the mean value of the pressure change p' in a 
sound wave. In the first approximation, corresponding to the usual linearised 
equations of motion, p' is a function which periodically changes sign, and the 
mean value of />' is zero. This result, however, ceases to hold if we go to 
higher approximations. If we take only second-order quantities, p' can be 
expressed in terms of quantities calculated from the linear sound equations, so 
that it is not necessary to solve directly the non-linear equations of motion 
obtained when terms of higher order are taken into account. 

We start from Bernoulli's equation : to + \v 2 + d<j>jdt = constant, and average 
it with respect to time. The mean value of the time derivative d$\dt is zero.f 
Putting also to = too + w' and including too in the constant, we obtain 
to' + \v 2 = constant. We suppose that the wave is propagated in an infinite 
volume of fluid but is damped at infinity, i.e. v, to', etc. are zero at infinity. 



t By the general definition of the mean value, we have for the mean derivative of any function /(*) 

T 



1 fd/ 

d//d* = lim — dt = hm 

•" r-*. 2T J dt r-»» 



f(T)-f{-T) 



IT 

T 
If the function /(*) remains finite for all t , the limit is zero, so that d//d* = 0. 



§65 Reflection and refraction of sound waves 253 

Since the constant is the same in all space, it must evidently be zero, so that 

^+1^2 = o. (64.9) 

We next expand zv' in powers of p', and take only the terms up to the second 
order : 

to' = {dw\dp) s p' +\{d 2 w\dp 2 ) s p' 2 \ 

since (ckvjdp) s = 1/p, we have 

p' p' 2 / dp \ p' p' 2 

w' = 



po 2po 2 \8p/ s po 2c 2 po 2 
Substituting this in (64.9) gives 

y= -Ip^+^Jlpoc 2 = -ipo^+ P^^/lpo, (64.10) 

which determines the required mean value. The expression on the right is a 
second-order quantity, and is calculated by using the />' and v obtained from 
the solution of the linearised equations of motion. The mean density is 

J' = {dp\dp*) s p' +\{d 2 p\dp* 2 ) s J 2 . (64.11) 

If the wave may be regarded as a travelling plane wave in the volume 
concerned, then v = cp'fpo, so that v 2 = c 2 p' 2 lpo 2 , and the expression 
(64.10) is zero, i.e. the mean pressure variation in a plane wave is an effect 
of higher order than the second. The density variation p = %(d 2 pldpo 2 ) a P' 2 
is not zero, however. (We may mention that the derivative (d 2 p[dpo 2 ) s is in 
fact always negative, and therefore p < in a travelling wave.) In the same 
approximation, we have for the mean value of the momentum flux density 
tensor in a travelling plane wave p8i]c+ pvivjc = po&ik + ptfViVjc. The first term 
is the equilibrium pressure and does not relate to the sound wave. In the 
second term, we introduce the unit vector n in the direction of v (the same 
as the direction of propagation of the wave), and, using (64.2), obtain for the 
momentum flux density in a sound wave 

U ik = Entn k . (64.12) 

If the wave is propagated in the ^-direction, only the component U X x = E 
is not zero. Thus, in this approximation, there is in the plane sound wave only 
an ^-component of the mean momentum flux, and this is transmitted in the 
^-direction. 

§65. Reflection and refraction of sound waves 

When a sound wave is incident on the boundary between two different fluid 
media, it undergoes reflection and refraction. This means that, in addition to 



254 Sound §65 

the incident wave, two more appear; one (the reflected wave) is propagated 
back into the first medium from the surface of separation, and the other (the 
refracted wave) is propagated into the second medium. Consequently, the 
motion in the first medium is a combination of two waves (the incident and 
the reflected), whereas in the second medium there is only one, the refracted 
wave. 

The relation between these three waves is determined by the boundary 
conditions at the surface of separation, which require the pressures and normal 
velocity components to be equal. 

Let us consider the reflection and refraction of a monochromatic longitudi- 
nal wave at a plane surface separating two media, which we take as the yz- 
plane. It is easy to see that all three waves have the same frequency co and 
the same components k v , k z of the wave vector, but not the same component 
k x perpendicular to the plane of separation. For, in an infinite homogeneous 
medium, a monochromatic wave with constant k and a> satisfies the equations 
of motion. The presence of a boundary introduces only some boundary con- 
ditions, which in the case considered apply at x = 0, i.e. do not depend on 
the time or on the co-ordinates y, z. Hence the dependence of the solution 
on t, y and z remains the same in all space and time, i.e. o>, ky, and k z are 
the same as in the incident wave. 

From this result we can immediately derive the relations which give the 
directions of propagation of the reflected and refracted waves. Let the plane 
of the incident wave be the ry-plane. Then k z = in the incident wave, and 
the same must be true of the reflected and refracted waves. Thus the direc- 
tions of propagation of the three waves are coplanar. 

Let 8 be the angle between the direction of propagation of the wave and 
the ar-axis. Then, from the equality of ky = (oi/c) sin for the incident 
and reflected waves, it follows that 

h = 0i', (65.1) 

i.e. the angle of incidence d\ is equal to the angle of reflection #i'. From a 
similar equation for the incident and refracted waves it follows that 

sin#i/sin02 = £1/^2. (65.2) 

which relates the angle of incidence Q\ to the angle of refraction 9% (c\ and c% 
being the velocities of sound in the two media). 

In order to obtain a quantitative relation between the intensities of the 
three waves, we write the respective velocity potentials as 

<f>i — Aiexp[ico{(x/ci) cosdi + (yJci) sindi — t}], 

<f>i = A\ exp[/ct){( — x\c\) cosdi + {yjci) sin0i — *}], 

<f> 2 = ^2exp[/cy{(«r/c2)cos^2 + (j/^2)sin^2— *}]• 

On the surface of separation (x = 0) the pressure (p — —pdtf>jdt) and the 



§65 Reflection and refraction of sound waves 255 

normal velocities (v x = d<f>ldx) in the two media must be equal; these con- 
ditions lead to the equations 

cos 6i M , v cos 62 m 
px{A x + A{) = p 2 A 2y {A x - At!) = A 2 . 

C\ C 2 

The reflection coefficient R is defined as the ratio of the (time) average energy 
flux densities in the reflected and incident waves. Since the energy flux 
density in a plane wave is cpv 2 , we have R = cipivi 2 lapivi 2 = |^4i'| 2 /|^i| 2 . 
A simple calculation gives 

\ />2 tan 02 + pi tan 0i / 

The angles 0\ and 2 are related by (65.2); expressing 2 in terms of 0i, we 
can put the reflection coefficient in the form 

f p 2 C 2 COS 0! - piVJC! 2 - C 2 2 sin 2 0^ 1 2 

[ p 2 c 2 cos 0! + piV( c i 2 - c 2 2 sin 2 0i) J 
For normal incidence (#i = 0), this formula gives simply 

R = (^Zf^)\ (65.5) 

\P2C2 + PlCl/ 

For an angle of incidence such that 

pl 2 (Cl 2 -C2 2 ) 

the reflection coefficient is zero, i.e. the wave is totally refracted. This can 
happen if c\ > C2 but p 2 c 2 > pici, or if both inequalities are reversed. 

PROBLEM 

Determine the pressure exerted by a sound wave on the boundary separating two fluids. 

Solution. The sum of the total energy fluxes in the reflected and refracted waves must 
equal the incident energy flux. Taking the energy flux per unit area of the surface of separa- 
tion, we can write this condition in the form CxE-^cos 0x = c^'cos ^ 1 +tr 2 JE , s cos Zt where 
E lt Ei and E t are the energy densities in the three waves. Introducing the reflection coefficient 
R = Ei/E u we therefore have 

— C\ cos 01 — 

C2 COS 02 

The required pressure p is determined as the ^-component of the momentum lost per unit 
time by the sound wave (per unit area of the boundary). Using the expression (64.12) for 
the momentum flux density tensor in a sound wave, we find 

p = Ei cosWx + Ek' cos 2 0!-E 2 cos 2 2 . 

Substituting for E 2 , introducing R and using (65.2), we obtain 

p = Ex sin 0i cos 0i[(l + R) cot 0i - (1 - R) cot 2 ]. 



256 Sound §66 

For normal incidence {6 1 = 0), we find, using (65.5), 

^ I" Pl 2 d 2 +P2 2 C2 2 - 2p 1 p 2 C 1 2 l 
L (j>lCl+p2C2) 2 J' 

§66. Geometrical acoustics 

A plane wave has the distinctive property that its direction of propagation 
and its amplitude are the same in all space. An arbitrary sound wave, of 
course, does not possess this property. However, cases can occur where a 
sound wave that is not plane may still be regarded as plane in any small 
region of space. For this to be so it is evidently necessary that the amplitude 
and the direction of propagation should vary only slightly over distances of 
the order of the wavelength. 

If this condition holds, we can introduce the idea of rays, these being lines 
such that the tangent to them at any point is in the same direction as the 
direction of propagation; and we can say that the sound is propagated along 
the rays, and ignore its wave nature. The study of the laws of propagation 
of sound in such cases is the task of geometrical acoustics. We may say that 
geometrical acoustics corresponds to the limit of small wavelengths, A -» 0. 

Let us derive the basic equation of geometrical acoustics, which determines 
the direction of the rays. We write the wave velocity potential as 

<f> = aeW. (66.1) 

In the case where the wave is not plane but geometrical acoustics can be 
applied, the amplitude a is a slowly varying function of the co-ordinates and 
the time, while the wave phase ift is "almost linear" (we recall that in a plane 
wave ifs = k»r— wt+a., with constant k and co). Over small regions of space 
and short intervals of time, the phase ip may be expanded in series; up to 
terms of the first order we have 

tfi = ifjo + r-gradip+tdipjdt. 

In accordance with the fact that, in any small region of space (and during 
short intervals of time), the wave may be regarded as plane, we define the 
wave vector and the frequency at each point as, 

k = dt/tjdr = grad«/r, co = -di/i/dt. (66.2) 

The quantity if/ is called the eikonal. 

In a sound wave we have w 2 jc 2 = k 2 = k x 2 +k y 2 + k g 2 . Substituting (66.2), 
we obtain the basic equation of geometrical acoustics : 

If the fluid is not homogeneous, the coefficient 1/c 2 is a function of the co- 
ordinates. 



§66 Geometrical acoustics 257 

As we know from mechanics, the motion of material particles can be 
determined by means of the Hamilton-Jacobi equation, which, like (66.3), 
is a first-order partial differential equation. The quantity analogous to ift 
is the action S of the particle, and the derivatives of the action determine the 
momentum p = dSfdr and the Hamilton's function (the energy) of the particle 
H = - dSjdt\ these formulae are similar to (66.2). We know, also, that the 
Hamilton-Jacobi equation is equivalent to Hamilton's equations 

p = - dH/dr, v = r = dH/dp. 

From the above analogy between the mechanics of a material particle and 
geometrical acoustics, we can write down similar equations for rays: 

k = -dcoldr, i = dco/dk. (66.4) 

In a homogeneous isotropic medium co = ck with c constant, so that k = 0, 
r = en (n being a unit vector in the direction of k), i.e. the rays are propagated 
in straight lines with a constant frequency to, as we should expect. 

The frequency, of course, remains constant along a ray in all cases where 
the propagation of sound occurs under steady conditions, i.e. the properties of 
the medium at each point in space do not vary with time. For the total time 
derivative of the frequency, which gives its rate of variation along a ray, is 
dcofdt = dcoldt + r-dcjjdr+k-dcoldk. On substituting (66.4), the last two 
terms cancel, and in a steady state dcofdt = 0, so that dcofdt = 0. 

In steady propagation of sound in an inhomogeneous medium at rest 
co = ck, where c is a given function of the co-ordinates. The equations 
(66.4) give 

r = en, k = -k grade. (66.5) 

The magnitude of the vector k varies along a ray simply according to k = cofc 
(with co constant). To determine the change in direction of n we put 
k = conic in the second of (66.5): con/e - (con/e 2 )(r . grade) = -k grade, 
whence dn/d* = -grade + n(n« grade). Introducing the element of length 
along the ray dl — c dt, we can rewrite this equation 

dn/d/ = - (1/e) grade + n(n-grade)/e. (66.6) 

This equation determines the form of the rays; n is a unit vector tangential 

to a ray.f 

If equation (66.3) is solved, and the eikonal iff is a known function of 
co-ordinates and time, we can then find also the distribution of sound inten- 
sity in space. In steady conditions, it is given by the equation div q = 
(q being the sound energy flux density), which must hold in all space except 



t As we know from differential geometry, the derivative dn/d/ along the ray is equal to N/R, where 
N is a unit vector along the principal normal and R is the radius of curvature of the ray. The expres- 
sion on the right-hand side of (66.6) is, apart from a factor 1/c, the derivative of the velocity of sound 
along the principal normal; hence we can write the equation as l/ic = -(l/c)N«grad c. The rays 
bend towards the region where c is smaller. 



258 Sound §66 

at sources of sound. Putting q = cEn, where E is the sound energy density 
(see (64.6)), and remembering that n is a unit vector in the direction of 
k = grad ifj, we obtain the equation 

6iv(cE grad 0/|grad 0|) = 0, (66.7) 

which determines the distribution of E in space. 

The second formula (66.4) gives the velocity of propagation of the waves 
from the known dependence of the frequency on the components of the 
wave vector. This is a very important formula, which holds not only for 
sound waves, but for all waves (for example, we have already applied it to 
gravity waves in §12). We shall give here another derivation of this formula, 
which puts in evidence the meaning of the velocity which it defines. Let us 
consider a wave packet, which occupies some finite region of space. We 
assume that its spectral composition includes monochromatic components 
whose frequencies lie in only a small range; the same is true of the compo- 
nents of their wave vectors. Let to be some mean frequency of the wave 
packet, and k a mean wave vector. Then, at some initial instant, the wave 
packet is described by a function of the form 

cf> = exp(zk.r)/(r). (66.8) 

The function /(r) is appreciably different from zero only in a region which is 
small (though it is large compared with the wavelength \jk). Its expansion as 
a Fourier integral contains, by the above assumptions, components of the 
form exp(av Ak), where Ak is small. 

Thus each monochromatic component is, at the initial instant, 

<£k = constantx exp[Y(k + Ak) • r] . (66.9) 

The corresponding frequency is <o(k + Ak) (we recall that the frequency is a 
function of the wave vector). Hence the same component at time t has the 
form 

<f> k = constantx exp[Y(k + Ak) • r — ico(k + Ak)*]. 

We use the fact that Ak is small, and expand w(k + Ak) in series, taking 
only the first twO terms: co(k + Ak) = <x) + (dco/8k)'^k, where co = co(k) is 
the frequency corresponding to the mean wave vector. Then <f> k becomes 

<f> k = constant x exp|>'(k- r - cot)] exp|YAk' (r - tdco/dk)]. (66. 10) 

If we now sum all the monochromatic components, with all the Ak that 
occur in the wave packet, we see from (66.9) and (66.10) that the result is 

<f> = exp[*(k • r - cot)]f(r - tdco/dk), (66.11) 

where/is the same function as in (66.8). A comparison with (66.8) shows that, 
after a time t, the amplitude distribution has moved as a whole through a 
distance tdcojdk; the exponential coefficient of /in (66.11) affects only the 
phase. Consequently, the velocity of the wave packet is 

U = dcojdk. (66.12) 



§67 Propagation of sound in a moving medium 259 

This formula gives the velocity of propagation for any dependence of co 
on k.f When a) = ck, with c constant, it of course gives the usual result 
U = ojjk = c. In general, when co(k) is an arbitrary function, the velocity 
of propagation is a function of the frequency, and the direction of propaga- 
tion may not be the same as that of the wave vector. 

PROBLEM 

Determine the altitude variation in the amplitude of sound propagated in an isothermal 
atmosphere under gravity. 

Solution. In an isothermal atmosphere (regarded as a perfect gas) the velocity of sound 
is constant. The energy flux density evidently decreases along a ray in inverse proportion 
to the square of the distance r from the source: cpv* ~ 1/r 2 . Hence it follows that the ampli- 
tude of the velocity fluctuations in the sound wave varies along a ray inversely as r\/ P ', according 
to the barometric formula, p ~ exp( - figz/RT), where z is the altitude, ft the molecular weight 
of the gas and R the gas constant. 

§67. Propagation of sound in a moving medium 

The relation w = ck between the frequency and the wave number is valid 
only for a monochromatic sound wave propagated in a medium at rest. It is 
not difficult to obtain a similar relation for a wave propagated in a moving 
medium (and observed in a fixed system of co-ordinates). 

Let us consider a homogeneous flow of velocity u. We take a fixed system 
K of co-ordinates x, y, z, and also a system K' of co-ordinates x\ y\ z' 
moving with velocity u relative to K. In the system K! the fluid is at rest, 
and a monochromatic wave has the usual form cf> = constant x exp[/(k«r' - kct)]. 
The radius vector r' in the system K' is related to the radius vector r in 
the system K by r' = r-ut. Hence, in the fixed system of co-ordinates, the 
wave has the form <j> = constant x exp{*[k«r - (kc + k»u)*]}. The coeffici- 
ent of t in the exponent is the frequency w of the wave. Thus the frequency 
in a moving medium is related to the wave vector k by 

a, = ck + wk. (67.1) 

The velocity of propagation is 

dco/dk = ck/k + u; (67.2) 

this is the vector sum of the velocity c in the direction of k and the velocity 
u with which the sound is "carried along" by the moving fluid. 

Using formula (67.1), we can investigate what is called the Doppler effect: 



t The velocity defined by (66.12) is called the group velocity of the wave, and the ratio ufk the 
phase velocity. However, it must be borne in mind that the phase velocity does not correspond to 
any actual physical propagation. 

Regarding the derivation given here it should be emphasised that the motion of the wave packet 
without change of form (i.e. without change in the spatial distribution of the amplitude), expressed 
by (66.11), is approximate, and results from the assumption that the range Ak is small. In general, 
when U depends on <a, a wave packet is "smoothed out" during its propagation, and the region of 
space which it occupies increases in size. It can be shown that the amount of this smoothing out 
is proportional to the squared magnitude of the range Ak of the wave vectors which occur in the 
composition of the wave packet. 



260 Sound §67 

the frequency of sound, as received by an observer moving relative to the 
source, is not the same as the frequency of oscillation of the source. 

Let sound emitted by a source at rest (relative to the medium) be received 
by an observer moving with velocity u. In a system K' at rest relative to the 
medium we have k = o> /c, where o> is the frequency of oscillation of the 
source. In a system K moving with the observer, the medium moves with 
velocity -u, and the frequency of the sound is, by (67.1), a) = ck-u>k. 
Introducing the angle 6 between the direction of the velocity u and that of 
the wave vector k, and putting k = wqJc, we find that the frequency of the 
sound received by the moving observer is 

at = co [l-(u/c) cos 6]. (67.3) 

The opposite case, to a certain extent, is the propagation in a medium at 
rest of a sound wave emitted from a moving source. Let u be now the velocity 
of the source. We change from the fixed system of co-ordinates to a system K' 
moving with the source; in the system K', the fluid moves with velocity -u. 
In K', where the source is at rest, the frequency of the emitted sound wave 
must equal the frequency a> of the oscillations of the source. Changing the 
sign of u in (67.1) and introducing the angle 6 between the directions of u 
and k, we have o> = ck[l-(u/c) cos ff]. In the original fixed system K, 
however, the frequency and the wave vector are related by to = ck. Thus we 
find 

w = a>o/[l-(ujc) cos 6]. (67.4) 

This formula gives the relation between the frequency co of the oscillations 
of a moving source and the frequency w of the sound heard by an observer 
at rest. 

If the source is moving away from the observer, the angle 6 between its 
velocity and the direction to the observer lies in the range \n < 6 ^ it, 
so that cos 6 < 0. It then follows from (67.4) that, if the source is moving 
away from the observer, the frequency of the sound heard is less than co . 

If, on the other hand, the source is approaching the observer, then 
^ 8 < \n, so that cos 6 > 0, and the frequency to > o> increases with 
u. For u cos 6 > c, according to formula (67.4) co becomes negative, which 
means that the sound heard by the observer actually reaches him in the 
reverse order, i.e. sound emitted by the source at any given instant arrives 
earlier than sound emitted at previous instants. 

As has been mentioned at the beginning of §66, the approximation of 
geometrical acoustics corresponds to the case of small wavelengths, i.e. large 
magnitudes of the wave vector. For this to be so the frequency of the sound 
must in general be large. In the acoustics of moving media, however, the 
latter condition need not be fulfilled if the velocity of the medium exceeds 
that of sound. For in this case k can be large even when the frequency is 
zero; from (67.1) we have for co = the equation 

ck= -u-k, (67.5) 



§67 Propagation of sound in a moving medium 261 

and this has solutions if u > c. Thus, in a medium moving with supersonic 
velocities, there can be steady small perturbations described (if k is sufficiently 
large) by geometrical acoustics. This means that such perturbations are 
propagated along rays. 

Let us consider, for example, a homogeneous supersonic stream moving 
with constant velocity u, whose direction we take as the #-axis. The vector 
k is taken to lie in the ry-plane, and its components are related by 

( M 2_ C 2)^2 = c 2 ky ^ (67.6) 

which is obtained by squaring both sides of equation (67.5). To determine the 
form of the rays, we use the equations of geometrical acoustics (66.4), 
according to which x = 8wldk x ,y = dcojdky. Dividing one of these equations 
by the other, we have dyjdx = (8a)l8ky)l(8o)l8k x ). This relation, however, 
is, by the rule of differentiation for implicit functions, just the derivative 
— dk x jdky taken at a constant frequency (in this case zero). Thus the equation 
which gives the form of the rays from the known relation between k x and k y is 

dyjdx = —dk x jdky. (67.7) 

Substituting (67.6), we obtain 

dyfdx = ±cJ-\/{u 2 — c 2 ). 

For constant u this equation represents two straight lines intersecting the 
#-axis at angles ± a, where sin a = cju. 

We shall return to a detailed study of these rays in gas dynamics, where they 
are very important; see in particular §§79, 96 and 109. 

PROBLEMS 

Problem 1. Derive an equation giving the form of sound rays propagated in a steadily 
moving homogeneous medium with a velocity distribution u(x, y, z), when u <^ c every- 
where, f 

Solution. Substituting (67.1) in (66.4), we obtain the equations of propagation of the 
rays in the form 

k = — (k'grad)u — kx curlu, r = v = ck/k + u. 

Using these equations, and also 

dufdt = du/dt + (v'grad)u = (vgrad)u a (c/£)(k«grad)u, 

we calculate the derivative d(kv)fdt, retaining only terms as far as the first order in u. The 
result is d(kv)ldt = —kvnX curlu, where n is a unit vector in the direction of v. But 
d(kv)/dt = nd(kv)/dt +kv dnfdt. Since n and dn/dt are perpendicular (because n 2 = 1, 
and therefore n«n = 0), it follows from the above equations that n = — nXcurl u. Intro- 
ducing the element of length along the ray dl = c dt, we can write finally 

dn/dl = - n X curl u/c. (1) 

This equation determines the form of the rays ; n is a unit tangential vector (and is no longer 
in the same direction as k). 



t It is assumed that the velocity u varies only over distances large compared with the wavelength 
of the sound. 



262 Sound §68 

Problem 2. Determine the form of sound rays in a moving medium with a velocity distri- 
bution u x = u(z), u y = u t = 0. 

Solution. Expanding equation (1), Problem 1, we find dn x /dl = (n z [c)du/dz, dn y /dl = 0; 
the equation for n z need not be written down, since n a = 1. The second equation gives 
n y = constant e= % )0 . In the first equation we write n z = dzjdl, and then we have by inte- 
gration n x = n Xt0 +u(z)lc These formulae give the required solution. 

Let us assume that the velocity u is zero for z = and increases upwards (du/dz > 0). 
If the sound is propagated "against the wind" (n x < 0), its path is curved upwards; if it is 
propagated "with the wind" (n x > 0), its path is curved downwards. In the latter case a 
ray leaving the point z = at a small angle to the x-axis (i.e. with n m , close to unity) rises 
only to a finite altitude z = 2 max , which can be calculated as follows. At the altitude z mAX 
the ray is horizontal, i.e. n z = 0. Hence we have 

»z 2 + % 2 « n Xt o 2 + nyfi 2 + 2n x oulc = 1, 

so that 2n x ,ou(z m& z)lc = w z ,o 2 , whence we can determine 2 max from the given function u(z) 
and the initial direction n of the ray. 

Problem 3. Obtain the expression of Fermat's principle for sound rays in a steadily moving 
medium. 

Solution. Fermat's principle is that the integral 



cfk-dl, 



taken along a ray between two given points, is a minimum; k is supposed expressed as a 
function of the frequency to and the direction n of the ray.f This function can be found by 
eliminating v and k from the relations <o = ck+wh. and vn = ck/k+u. Fermat's principle 
then takes the form 

S^{V[(t 2 -«2)d/2+(u.dl)2]-U.dl}/(c2- tt 2) = 0. 
In a medium at rest, this integral reduces to the usual one, j dljc. 

§68. Characteristic vibrations 

Hitherto we have discussed only oscillatory motion in infinite media, and 
we have seen, in particular, that in such media waves of any frequency can be 
propagated. 

The situation is very different when we consider a fluid in a vessel of finite 
dimensions. The equations of motion themselves (the wave equations) are 
of course unchanged, but they must now be supplemented by boundary 
conditions to be satisfied at the solid walls or at the free surface of the fluid. 
We shall consider here only what are called free vibrations, i.e. those which 
occur in the absence of variable external forces. Vibrations occurring as a 
result of external forces are called forced vibrations. 

The equations of motion for a finite fluid do not have solutions satisfying 
the appropriate boundary conditions for every frequency. Such solutions 
exist only for a series of definite frequencies <o. In other words, in a medium 
of finite volume, free vibrations can occur only with certain frequencies. 
These are called the characteristic frequencies of the fluid in the vessel 
concerned. 

The actual values of the characteristic frequencies depend on the size and 



f See The Classical Theory of Fields, §7-1, Addison- Wesley Press, Cambridge (Mass.) 1951. 



§68 Characteristic vibrations 263 

shape of the vessel. In any given case there is an infinite number of charac- 
teristic frequencies. To find them, it is necessary to examine the equations 
of motion with the appropriate boundary conditions. 

The order of magnitude of the first (i.e. smallest) characteristic frequency 
can be seen at once from dimensional considerations. The only parameter 
having the dimensions of length which appears in the problem is the linear 
dimension / of the body. Hence it is clear that the wavelength Ai correspond- 
ing to the first characteristic frequency must be of the order of /, and the order 
of magnitude of the frequency cui itself is obtained by dividing the velocity 
of sound by the wavelength. Thus 

Ai - /, oil ~ c\l. (68.1) 

Let us ascertain the nature of the motion in characteristic vibrations. 
If we seek a solution of the wave equation for the velocity potential (say) 
which is periodic in time, of the form <j> — <j>q{x, y, z)e~ i<ot , then we have for 
<f>o the equation 

A<h + («> 2 lc 2 )<h = 0. (68.2) 

In an infinite medium, where no boundary conditions need be applied, this 
equation has both real and complex solutions. In particular, it has a solution 
proportional to e ik ' T , which gives a velocity potential of the form 

<j> = constant xexp[i(k'r—cot)]. 

Such a solution represents a wave propagated with a definite velocity — a 
travelling wave. 

For a medium of finite volume, however, complex solutions cannot in 
general exist. This can be seen as follows. The equation satisfied by <f>Q 
is real, and the boundary conditions are real also. Hence, if <f>o(x, y, z) is a 
solution of the equations of motion, the complex conjugate function <f>o* 
is also a solution. Since, however, the solution of the equations for given 
boundary conditions is in general uniquef apart from a constant factor, we 
must have <£o* = constant x <£o, where the constant is complex and its 
modulus is clearly unity. Thus <f>o must be of the form <f>o = f(x, y, z)e~ ia -, 
the function / and the constant a being real. The potential <f> is thus of the 
form (taking the real part of (f>oe~ iwt ) 

<f> = f{ x >y> % ) cos(cot + <x), (68.3) 

i.e. it is the product of some function of the co-ordinates and a simple periodic 
function of the time. 

This solution has properties entirely different from those of a travelling 
wave. In the latter, where <f> = constant xcos(k«r— «j^ + a), the phase 
k«r — cot +<x of the oscillations at different points in space is different at any 
given instant, except only at points for which k • r differs by an integral 



f This may not be true when the vessel is highly symmetrical in form (e.g. a sphere). 



264 Sound §68 

multiple of the wavelength. In the wave represented by (68.3), all points are 
oscillating in the same phase tot + at. at any given instant. Such a wave is 
obviously not "propagated" ; it is called a stationary wave. Thus the charac- 
teristic vibrations are stationary waves. 

Let us consider a stationary plane sound wave, in which all quantities are 
functions of one co-ordinate only (x, say) and of time. Writing the general 
solution of 8 2 <f>ojdx 2 + 2 co(f)olc 2 = in the form <f>o = a cos(a»*/c+/S), we have 
<f> = a cos(cor+a) cos(cox/c+l3). By an appropriate choice of the origins 
of x and t, we can make a and /S zero, so that 

<f> = a cos cot cos cox/c. (68.4) 

For the velocity and pressure in the wave we have 

v = 8(f>Jdx = — (aco/c) cos cot sin cox/c ; 
p' = —p 8<f>/8t = pco sin cot cos cox/c. 
At the points x = 0, ire/ to, 2,-nc/to, ..., which are at a distance ttc/co = £A 
apart, the velocity v is always zero ; these points are called nodes of the velocity. 
The points midway between them (x = ttc/Ico, Zttc/2co, ...) are those at 
which the amplitude of the time variations of the velocity is greatest. These 
are called antinodes. The pressure p' evidently has nodes and antinodes in 
the reverse positions. Thus, in a stationary plane wave, the nodes of the 
pressure are the antinodes of the velocity, and vice versa. 

An interesting case of characteristic vibrations is that of the vibrations of 
a gas in a vessel having a small aperture (a resonator). In a closed vessel the 
smallest characteristic frequency is, as we know, of the order of c/l, where / 
is the linear dimension of the vessel. When there is a small aperture, however, 
new characteristic vibrations of considerably smaller frequency appear. 
These are due to the fact that, if there is a pressure difference between the 
gas in the vessel and that outside, this difference can be equalised by the 
motion of gas into or out of the vessel. Thus oscillations appear which 
involve an exchange of gas between the resonator and the outside medium. 
Since the aperture is small, this exchange takes place only slowly, and hence 
the period of the oscillations is large, and the frequency correspondingly 
small (see Problem 2). The frequencies of the ordinary vibrations occurring 
in a closed vessel are practically unchanged by the presence of a small aper- 
ture. 

PROBLEMS 

Problem 1 . Determine the characteristic frequencies of sound waves in a fluid contained 
in a cuboidal vessel. 

Solution. We seek a solution of the equation (68.2) in the form 
</>q = constant x cos qx cos ry cos sz, 

where q 2 +r i +s* = w 2 /c 2 . At the walls of the vessel we have the conditions v x = d<f>/dx = 
for x = and a, d<f>[dy = for y = and b, 8<f>/dz = for z = and c, where a, b, c are 
the sides of the cuboid. Hence we find q — m-nla, r = nn/b, s = p-n\c, where m, n, p are 
any integers. Thus the characteristic frequencies are 

co 2 = c 2 7T 2 (m 2 /a 2 +n 2 /b 2 +p 2 /c 2 ). 



§69 Spherical waves 265 

Problem 2. A narrow tube of cross-sectional area S and length / is fixed to the aperture of 
a resonator. Determine the characteristic frequency. 

Solution. Since the tube is narrow, in considering oscillations accompanied by the 
movement of gas into and out of the resonator we can suppose that only the gas in the tube 
has an appreciable velocity, while the gas in the vessel is almost at rest. The mass of gas in 
the tube is Spl, and the force on it is S(p —p), where p and p are the gas pressures inside and 
outside the resonator respectively. Hence we must have Spiv = S(p—p ), where v is the 
gas velocity in the tube. The time derivative of the pressure is given by p = c 2 p, and the 
decrease per unit time in the gas density in the resonator ( — p) can be supposed equal to the 
mass of gas leaving the resonator per unit time (Spv) divided by the volume V of the resonator. 
Thus we have£ «= —c^SpvfV, whence 

/>"= -c 2 SpvjV = -c*S(p-p )llV. 

This equation gives p—po = constant X cos to t, where the characteristic frequency 
(o — c\Z{SjlV). This is small compared with cjL (where L is the linear dimension of the 
vessel), and the wavelength is therefore large compared with L. 

In solving this problem we have supposed that the linear amplitude of the oscillations of 
gas in the tube is small compared with its length /. If this were not so, the oscillations would 
be accompanied by the outflow of a considerable fraction of the gas in the tube, and the linear 
equation of motion used above would be inapplicable. 

§69. Spherical waves 

Let us consider a sound wave in which the distribution of density, velocity, 
etc., depends only on the distance from some point, i.e. is spherically sym- 
metrical. Such a wave is called a spherical wave. 

Let us determine the general solution of the wave equation which represents 
a spherical wave. We take the wave equation for the velocity potential: 
/\<f> — (llc 2 )d 2 <f>jdt 2 = 0. Since ^ is a function only of the distance r from the 
centre and of the time t y we have, using the expression for the Laplacian in 
spherical co-ordinates, 

8 2 (f> 1 d l 86 \ 

-Z = C 2 ( r 2_Z . (69.1) 

dt 2 r 2 dr \ dr J 

We seek a solution in the form <f> = /(r, t)jr. Substituting, we have after 
a simple calculation the following equation for /: d 2 fjdt 2 = c 2 d 2 fjdr 2 . This is 
just the ordinary one-dimensional wave equation, with the radius r as 
the co-ordinate. The solution of this equation is, as we know, of the form 
f = fi(ct—r)+f2(ct+r), where /i and f 2 are arbitrary functions. Thus the 
general solution of equation (69.1) is of the form 

^.^ =! ) + ^+o (69 . 2) 

r r 

The first term is an outgoing wave, propagated in all directions from the origin. 
The second term is a wave coming in to the centre. Unlike a plane wave, 
whose amplitude remains constant, a spherical wave has an amplitude which 
decreases inversely as the distance from the centre. The intensity in the wave 
is given by the square of the amplitude, and falls off inversely as the square of 
the distance, as it should, since the total energy flux in the wave is distributed 
over a surface whose area increases as r 2 . 



266 Sound §69 

The variable parts of the pressure and density are related to the potential 
by p' = —pd(j>jdt, p = —{pjc 2 )d<j>Jdt, and their distribution is determined by 
formulae of the same form as (69.2). The (radial) velocity distribution, how- 
ever, being given by the gradient of the potential, is of the form 

v = y- f) ; /a(rf+f) ). (69.3) 

If there is no source of sound at the origin, the potential (69.2) must remain 
finite for r = 0. For this to be so we must have/i(c£) = —fact), i.e. <f> is 
of the form 

f(ct-r)-f(ct+r) 

<f> = (69.4) 

r 

(a stationary spherical wave). If there is a source at the origin, on the other 
hand, the potential of the outgoing wave from it is <f> = f(ct—r)/r; it need 
not remain finite at r = 0, since the solution holds only for the region 
outside sources. 

A monochromatic stationary spherical wave is of the form 

sinkr 

<£ = Ae~M , (69.5) 

r 

where k = wjc. An outgoing monochromatic spherical wave is given by 

<f> = Ae^-^lr. (69.6) 

It is useful to note that this expression satisfies the differential equation 

A<f> + &<j> = -^rAe-^B(r), (69.7) 

where on the right-hand side we have the delta function S(r) = 8(x)B(y)8(z). 
For S(r) = everywhere except at the origin, and we return to the homo- 
geneous equation (69.1); and, integrating (69.7) over the volume of a small 
sphere including the origin (where the expression (69.6) reduces to Ae^^/r) 
we obtain — 4irAe~ i<ot on each side. 

Let us consider an outgoing spherical wave, occupying a spherical shell 
outside which the medium is either at rest or very nearly so ; such a wave can 
originate from a source which emits during a finite interval of time only, or 
from some region where there is a sound disturbance (cf. the end of §71, 
and §73, Problem 4). Before the wave arrives at any given point, the potential 
is <j> = 0. After the wave has passed, the motion must die away; this means 
that <f> must become constant. In an outgoing spherical wave, however, the 
potential is a function of the form <j> = f(ct — r)/r; such a function can tend 
to a constant only if the function / is zero identically. Thus the potential 
must be zero both before and after the passage of the wave.f From this we 



t Unlike what happens for a plane wave, after whose passage we can have ^ = constant =£ 



§69 Spherical waves 267 

can draw an important conclusion concerning the distribution of conden- 
sations and rarefactions in a spherical wave. 

The variation of pressure in the wave is related to the potential by 
p' = — pd<f}{dt. From what has been said above, it is clear that, if we integrate 
p' over all time for a given r, the result is zero : 

00 

JVd* = 0. (69.8) 

—00 

This means that, as the spherical wave passes through a given point, both 
condensations (p' > 0) and rarefactions (/>' < 0) will be observed at that 
point. In this respect a spherical wave is markedly different from a plane 
wave, which may consist of condensations or rarefactions only. 

A similar pattern will be observed if we consider the manner of variation of 
p' with distance at a given instant; instead of the integral (69.8) we now 
consider another which also vanishes, namely 

00 

jrp'dr = 0. (69.9) 

o 

PROBLEMS 

Problem 1. At the initial instant, the gas inside a sphere of radius a is compressed so that 
p' = constant = A; outside this sphere, p' = 0. The initial velocity is zero in all space. 
Determine the subsequent motion. 

Solution. The initial conditions on the potential are <f> — for t = 0, and r < a or 
r > a; <j> = F(r) for t = 0, where F(r) = for r > a and F(r) = -c 2 A/p for r < a. We 
seek $ in the form 

xi * f{ct-r)-f{ct + r) 

r 

From the initial conditions we obtain /( — r) — /(r) = 0,f'(—r)—f'(r) = rF(r)jc. From the 
first equation we have f'(—r)+f'(r) = 0, which together with the second equation gives 
/'(r) = —/'(—*)— —rF(r)/2c. Finally, substituting the value of F(r), we find the following 
expressions for the derivative /'(f) and the function /(f) itself: 

for ||| > a, /'(*) = 0, fit) = 0; 

for III < a, /'(|) = cZbfo, /(|) = <|2_ a 2 )A/4/>j 

which give the solution of the problem. If we consider a point with r > a, i.e. outside the 
region of the initial compression, we have for the density 

for t < (r — a)jc, p = 0; 

for (r-a)/c < t < (r + a)jc, p = %(r-ct)&{r; 

for t > (r+a)lc, p = 0. 

The wave passes the point considered during a time interval 2a/c; in other words, the wave 
has the form of a spherical shell of thickness 2a, which at time t lies between the spheres of 
radii ct—a and ct+a. Within this shell the density varies linearly; in the outer part (r > ct), 
the gas is compressed (p' > 0), while in the inner part (r < ct) it is rarefied (p' < 0). 



268 Sound §70 

Problem 2. Determine the characteristic frequencies of centrally symmetrical sound 
oscillations in a spherical vessel of radius a. 

Solution. From the boundary condition d<j>jdr = f or r = a (where <f> is given by (69.5)) 
we find tan ka = ha, which determines the characteristic frequencies. The first (lowest) 
frequency is a>x = 4*49 c\a. 

§70. Cylindrical waves 

Let us now consider a wave in which the distribution of all quantities is 
homogeneous in some direction (which we take as the ar-axis) and has com- 
plete axial symmetry about that direction. This is called a cylindrical wave, 
and in it we have <f> = <f>(R, t), where R denotes the distance from the #-axis. 
Let us determine the general form of such an axisymmetric solution of the 
wave equation. This can be done by starting from the general spherically 
symmetrical solution (69.2). R is related to r by r 2 = R 2 + z 2 , so that <f> as 
given by formula (69.2) depends on z when R and t are given. A function 
which depends on R and t only and still satisfies the wave equation can be 
obtained by integrating (69.2) over all z from — oo to oo, or equally well 
from to oo. We can convert the integration over z to one over r. Since 
z = -\/(r 2 -i? 2 ), dz = r drj<\/(r 2 — R. 2 ). When z varies from to oo, r 
varies from R to oo. Hence we find the general axisymmetric solution to be 



r fi(ct-r) r fa(ct + r) t 
j, = jL± f_dr+ — —dr y (70.1) 



V(r*-R 2 ) J ^/(r 2 -R 2 ) 



where f\ and fa are arbitrary functions. The first term is an outgoing cylin- 
drical wave, and the second an ingoing one. 

Substituting in these integrals ct±r — g, we can rewrite formula (70.1) as 



ct-R 



(• m* + f mw , (702) 

-oo ct+R 

We see that the value of the potential at time t in the outgoing cylindrical 
wave is determined by the values of fa at times from — oo to t — Rjc; 
similarly, the values of fa which affect the ingoing wave are those at times 
from t + RIc to infinity. 

As in the spherical case, stationary waves are obtained when/i(£) = —fa(£). 
It can be shown that a stationary cylindrical wave can also be represented in 
the form 



ct+R 



9 J V[R 2 -(t-ctf] ( ' 

ct-R 

where F(g) is another arbitrary function. 



§70 Cylindrical waves 269 

Let us derive an expression for the potential in a monochromatic cylindrical 
wave. The wave equation for the potential <f)(R, t) in cylindrical co-ordinates 
is 



R 8R\ 8R/ c* 



8U 
— = 0. 

dp 



In a monochromatic wave <f> = e-* w< /(i?), and we have for the function /(i?) 
the equation/" +/'/!?+ A 2 / = 0. This is Bessel's equation of order zero. 
In a stationary cylindrical wave, <f> must remain finite for R = 0; the appro- 
priate solution isJo(kR), where Jo is a Bessel function of the first kind. Thus, 
in a stationary cylindrical wave, 

<f> = Ae-i»J (kR). (70.4) 

For R — the function Jo tends to unity, so that the amplitude tends to the 
finite limit A. At large distances R, Jo may be replaced by its asymptotic 
expression, and <£ then takes the form 

<k = A / i-V-***. (70.5) 

The solution corresponding to a monochromatic outgoing travelling wave is 

<£ = Ae-***H<P(kR) y (70.6) 

where Ho {1) is the Hankel function of the first kind, of order zero. For 
R -> this function has a logarithmic singularity: 

<f> £ (2^/tt) log(£i?)<?-K (70.7) 

At large distances we have the asymptotic formula 

i2exp[i(kR-a>t~-frr)] 

V(kR) 

We see that the amplitude of a cylindrical wave diminishes (at large distances) 
inversely as the square root of the distance from the axis, and the intensity 
therefore decreases as ljR. This result is obvious, since the total energy 
flux is distributed over a cylindrical surface, whose area increases propor- 
tionally to R as the wave is propagated. 

A cylindrical outgoing wave differs from a spherical or plane wave in the 
important respect that it has a forward front but no backward front: once the 
sound disturbance has reached a given point, it does not cease, but diminishes 
comparatively slowly as t -> oo. Suppose that the function /i(£) in the 
first term of (70.2) is different from zero only in some finite range 
h < £ < h- Then, at times such that ct > R+ &> we have 






S2 



Mm 






270 Sound §71 

As t -> oo, this expression tends to zero as 



= Vt j* 1 ® 6 ** 



Si 

i.e. inversely as the time. 

Thus the potential in an outgoing cylindrical wave, due to a source which 
operates only for a finite time, vanishes, though slowly, as t -> oo. This 
means that, as in the spherical case, the integral of p' over all time is zero: 



00 

fp'dt = 0. (70.9) 

—oo 

Hence a cylindrical wave, like a spherical wave, must necessarily include both 
condensations and rarefactions. 

§71. The general solution of the wave equation 

We shall now derive a general formula giving the solution of the wave 
equation in an infinite fluid for any initial conditions, i.e. giving the velocity 
and pressure distribution in the fluid at any instant in terms of their initial 
distribution. 

We first obtain some auxiliary formulae. Let <f>(x, y, z, t) and i[j(x, y, z, t) 
be any two solutions of the wave equation which vanish at infinity. We 
consider the integral 

/=/ («^-#)dF, 

taken over all space, and calculate its time derivative. Since <f> and ifj satisfy 
the equations A<f>-$lc 2 = and A«A~$/c 2 = 0, we have 

dl/dt = j {<f4-^)dV = c* j (M«A-M<£)dF 

= c 2 J div(<£ grad ifj — if/ grad <f>)dV. 

The last integral can be transformed into an integral over an infinitely distant 
surface, and is therefore zero. Thus we conclude that dljdt = 0, i.e. J is 
independent of time : 

f {<j4 - #)d V = constant. (71.1) 

Next, let us consider the following particular solution of the wave 
equation : 

4> = 8[r-c{to-t)]lr (71.2) 

(where r is the distance from some given point O, to is some definite instant, 



§71 The general solution of the wave equation 271 

and 8 denotes the delta function), and calculate the integral of ifj over all 
space. We have 

00 00 

j l fsdV = J* if*-4irr 2 dr = 4tt j rS[r-c(t -t)]dr. 
o o 

The argument of the delta function is zero for r = c(to — t) (we assume that 
*o > *)• Hence, from the properties of the delta function, we find 

jt/,dV = +nc(to-t). (71.3) 

Differentiating this equation with respect to time, we obtain 

ffdV = -4ttc. (71.4) 

We now substitute for «/r, in the integral (71.1), the function (71.2), and 
take <f> to be the required general solution of the wave equation. According 
to (71.1), J is a constant; using this, we write down the expressions for / 
at the instants t = and t = to, and equate the two. For t = t o the two 
functions t[t and if, are each different from zero only for r = 0. Hence, on 
integrating, we can put r = in (f> and (f> (i.e. take their values at the point O), 
and take <f> and <f> outside the integral: 

I = <f>{x,y, z y t ) j 4 &V- <f>{x,y, z, t ) j tfi dV, 

where x t y, z are the co-ordinates of O. According to (71.3) and (71.4), 
the second term is zero for t — to, and the first term gives 

I = - 47TC(j}(x > y, z y to). 

Let us now calculate /for t = 0. Putting = dif/Jdt = —dtpjdtoy and 
denoting by <£o the value of the function <f> for t = 0, we have 

/= _ (L^ + ^AdV^ --£- (foiftt-odV- f^MidF. 

We write the element of volume as dV = r 2 drdo, where do is an element of 
solid angle, and then we obtain, by the properties of the delta function, 

J foifft-odV = j cf>orh(r-cto)drdo = ct j <f>Q r = cU do\ 

the integral of <£o«A is similar. Thus 

a 



dt 



(cto <£o,r-ct<,do) — Ct <f>0,r=ct do. 



272 Sound §71 

Finally, equating the two expressions for / and omitting the suffix zero in to, 
we obtain 

<l>(x,y, *> = 7- -7-C h,r=ct do) + 1\ <f> 0t r==ct do . (71.5) 

This formula, called Poisson's formula, gives the spatial distribution of 
the potential at any instant in terms of the distribution of the potential and 
its time derivative (or, equivalently, in terms of the velocity and pressure 
distribution) at some initial instant. We see that the value of the potential 
at time t is determined by the values of <j> and (j> at time t = on the surface 
of a sphere centred at O, of radius ct. 

Let us suppose that, at the initial instant, <f>o and <j>o are different from zero 
only in some finite region of space, bounded by a closed surface C (Fig. 34). 




Fig. 34 



We consider the values of <f> at subsequent instants at some point O. These 
values are determined by the values of <£o and <£o at a distance ct from O. 
The spheres of radius ct pass through the region within the surface C only 
for djc < t ^ Djc, where d and D are the least and greatest distances from 
the point O to the surface C. At other instants, the integrands in (71.5) 
are zero. Thus the motion at O begins at time t = djc and ceases at time 
t = Djc. The wave propagated from the region inside C has a forward 
front and a backward front. The motion begins when the forward front 
arrives at the point in question, while on the backward front particles pre- 
viously oscillating come to rest. 

PROBLEM 

Derive the formula giving the potential in terms of the initial conditions for a wave depend- 
ing on only two co-ordinates, * and y. 

Solution. An element of area of a sphere of radius ct can be written d/ = c 2 t 2 do, where do 
is an element of solid angle. The projection of d/ on the xy-plane i s d* dy = df-\/[(ct) 2 — p s ]fct, 
where p is the distance of the point x, y from the centre of the sphere. Comparing the two 
expressions, we can write do = da: dy/c£\/[(c£) 2 ~P 2 ]- Denoting by x, y the co-ordinates of 
the point where we seek the value of ^, and by i, 17 the co-ordinates of a variable point in the 
region of integration, we can therefore replace do in the general formula (71.5) by 
d£ dt]/ct\/[(ct) 2 — (x— i) 2 — (y— i?) 2 ], doubling the resulting expression because d* dy is the 



§72 



The lateral wave 



273 



projection of two elements of area on opposite sides of the #y-plane. Thus 

V[(ct) 2 -(x-o 2 -(y-v) 2 ] 



IttC J J 



+ 



^o(£*?)d£d7? 



2ncJJ V[(ct) 2 -(x-0 2 -(y-n) 2 ]' 

where the integration is over a circle centred at O, of radius ct. If <j> and <f> are zero except 
in a finite region C of the xy-plane (or, more exactly, except in a cylindrical region with its 
generators parallel to the s-axis), the oscillations at the point O (Fig. 34) begin at time 
t = djc, where d is the least distance from O to a point in the region. After this time, however, 
circles of radius ct > d centred at O will always enclose part or all of the region C, and <f> 
will tend only asymptotically to zero. Thus, unlike three-dimensional waves, the two- 
dimensional waves here considered have a forward front but no backward front (cf. §70). 



Reflected wave 





Fig. 35 



§72. The lateral wave 

The reflection of a spherical wave from the surface separating two media 
is of particular interest in that it may be accompanied by an unusual pheno- 
menon, the appearance of a lateral wave. 

Let Q (Fig. 35) be the source of a spherical sound wave in medium 1, at a 



274 Sound §72 

distance / from the infinite plane surface separating media 1 and 2. The 
distance / is arbitrary, and need not be large compared with the wavelength A. 
Let the densities of the two media be pi, pz, and the velocities of sound in 
them ci, cz. We suppose first that c\ > ci ; then, at distances from the source 
large compared with A, the motion in medium 1 will be a superposition of 
two outgoing waves. One of these is the spherical wave emitted by the source 
(the direct wave) ; its potential is 

<£i° = eMr/r, (72.1) 

where r is the distance from the source, and the amplitude is arbitrarily taken 
to be unity. We shall, for brevity, omit the factor e~ t<ot from all expressions 
in the present section. 




Fig. 36 

The wave surfaces of the second (reflected) wave are spheres centred at Q\ 
the image of the source Q in the plane of separation; this is the locus of 
points P reached at a given time by rays which leave Q simultaneously and 
are reflected from the plane in accordance with the laws of geometrical acou- 
stics (in Fig. 36, the ray QAP with angles of incidence and reflection 6 is 
shown). The amplitude of the reflected wave decreases inversely as the 
distance r' from the point Q' (which is sometimes called an imaginary 
source), but depends also on the angle 6, as if each ray were reflected with the 
coefficient corresponding to the reflection of a plane wave at the given angle 
of incidence 6. In other words, at large distances the reflected wave is given 
by the formula 

, = *** r p 2g2 - m V( gl 2- g2 2 sin 2g) -| 

r' Lp2t2 + piV(^i 2 -^ 2 sin26l)J' K ' j 

cf. formula (65.4) for the reflection coefficient for a plane wave. This formula, 
which is clearly valid for large r', can be rigorously derived by the method 
shown below. 

A more interesting case is that where c\ < c<l. Here, besides the ordinary 
reflected wave (72.2), another wave appears in the first medium. The 



§72 The lateral wave 275 

chief properties of this wave can be seen from the following simple con- 
siderations. 

The ordinary reflected ray QAP (Fig. 36) obeys Fermat's principle in the 
sense that it is the quickest path from Q to P, among paths lying entirely in 
medium 1 and involving a single reflection. When c\ < C2, however, Fermat's 
principle is also satisfied by another path, where the ray is incident on the 
boundary at the critical angle of total internal reflection #o (sin #o = ^1/^2), 
then is propagated in medium 2 along the boundary, and finally returns to 
medium 1 at the angle do. The path is QBCP in Fig. 36, and it is evident 
that 9 > 6q. It is easy to see that this path also has the extremal property: 
the time taken to traverse it is less than for any other path from Q to P lying 
partly in medium 2. 

The geometrical locus of points P reached at the same time by rays which 
simultaneously leave Q along the path QB, and then return to medium 1 at 
various points C, is evidently a conical surface whose generators are perpen- 
dicular to lines drawn from the imaginary source Q' at an angle 6q. 

Thus, if c\ < C2, together with the ordinary reflected wave, which has 
a spherical front, there is propagated in medium 1 another wave, which has a 
conical front extending from the plane of separation (where it meets the 
refracted wave front in medium 2) to the point where it touches the spherical 
front of the reflected wave ; this occurs along the line of intersection with a 
cone of semi-angle #o and axis QQ' (Fig. 35). This conical wave is called the 
lateral wave. 

It is easy to see by a simple calculation that the time along the path QBCP 
(Fig. 36) is less than along the path QAP to the same point P. This means 
that a sound signal from the source Q reaches an observer at P first as the 
lateral wave, and only later as the ordinary reflected wave. 

It must be borne in mind that the lateral wave is an effect of wave acoustics, 
despite the fact that it allows the above simple interpretation in terms of the 
concepts of geometrical acoustics. We shall see below that the amplitude of 
the lateral wave tends to zero in the limit A -> 0. 

Let us now make a quantitative calculation. The propagation of a mono- 
chromatic sound wave from a point source is described by equation (69.7) : 

A<f> + k*<f> = -47rS(r-l), (72.3) 

where k = wjc and 1 is the radius vector of the source. The coefficient 
of the delta function is chosen so that the direct wave has the form (72.1). 
In what follows we take a system of co-ordinates with the ry-plane as the 
plane of separation and the #-axis along QQ', with the first medium in z > 0. 
At the surface of separation the pressure and the ^-component of the velocity, 
or (equivalently) p<f> and d(f>/dz, must be continuous. 

Using the general Fourier method, we obtain the solution in the form 

co 00 

<f> = j j <f> K (z) txp[i( Kx x+ Ky y)] &K X &Ky, (72.4) 



276 Sound §72 

where 

1 00 00 

^*) = 4^2 J J ^expt-'Va^+^Ky)]^^. ( 72 - 5 ) 

—oo — oo 

From the symmetry relative to the xy-plane it is evident that <£ K can depend 
only on the quantity \k\ = \/(k x 2 + k v 2 ). Using the well-known formula 

1 r 
Jo(u) = — cos(w sin <£)d<£, 
2tt J 
o 

we can therefore write (72.4) as 

<f> = 2tt j<f> K (z)J (KR) K dK, (72.6) 

o 

where R = -\/(x 2 +y 2 ) is the cylindrical co-ordinate (the distance from the 
jsr-axis). It is convenient for the subsequent calculations to transform this 
formula into one in which the integral is taken from — oo to oo, expressing 
the integrand in terms of the Hankel function Hq {1 \u). The latter has a 
logarithmic singularity at the point u = 0; if we agree to go from positive 
to negative real u by passing above the point u = in the complex w-plane, 
then Ho (1) (-u) = H ^\ue in ) = Ho (1 \u)-2J (u). Using this relation, we 
can rewrite (72.6) as 

<f> = 7t J <f> K (z)Ho a \KR) K die. (72.7) 

—00 

From equation (72.3) we find for the function j> K the equation 



d*2 



/ co 2 \ 1 

-(k 2 --)^= --8(*-/). (72.8) 



The delta function on the right-hand side of the equation can be eliminated 
by imposing on the function <f> K (z) (satisfying the homogeneous equation) 
the boundary conditions at z = I: 

[d^/d^-[d^/d^_= -1/77. 

The boundary conditions at z = are 

l>^]o+-[>^]o- = 0, 
[d<^d*]o + -[d4/d*]o- = 0. 



§72 The lateral wave 277 

We seek a solution in the form 

<f> K = Ae-W for z > I, \ 

<f> K = Be-^ + Ce^ for / > z > 0, [ (72.11) 

<f> K = De^ z for > z. J 

Here 

^2 = K 2-k^, [M2 2 = K 2 -k 2 2 (ki = Qi\c\, k 2 = C0/C 2 ), 

and we must put 

u, = +VU 2 -A 2 )for k > k, 
^ VV } (72.12) 

/x = — i-\/(k 2 — k 2 ) for k < k. 

The first of these is necessary so that <j> should not increase without limit as 
z -> oo, and the second so that <f> should represent an outgoing wave. The 
conditions (72.9) and (72.10) give four equations which determine the co- 
efficients A, B, C and D. A simple calculation gives 



B = c ^w-pm c = e l/l1 

D=C ^1_, A = B + Ce 2 ^. 

/*lp2 + )U,2pi 



(72.13) 



For p2 = pi, c 2 = c\ (i.e. when all space is occupied by one medium), 
B is zero and A = Ce 21 ^ ; the corresponding term in <f> is evidently the dire ct 
wave (72.1), and the reflected wave in which we are interested is therefore 

oo 

cf>i' = tt j B(K)e-^W»(KR)KdK. (72.14) 

—oo 

In this expression the path of integration has to be specified. It passes 
above the singular point k = (in the complex /c-plane), as we have already 
mentioned. The integrand also has singular points (branch points) at 
k = ±ki, ± k 2 , where m or \i 2 vanishes. In accordance with the conditions 
(72.10), the contour must pass below the points +ki, +k 2 , and above the 
points — ki, —k 2 . 

Let us investigate the resulting expression for large distances from the 
source. Replacing the Hankel function by its well-known asymptotic 
expression, we obtain 

<£i' = f f " lp2 ~^ 2pl /JL^pi-fr+^ + iKRidK. (72.15) 
J M^1P2 + f*2pl) N Li-nR 

Fig. 37 shows the path of integration C for the case c\ > c%. The integral can 
be calculated by means of the saddle-point method. The exponent 
i[(z + l)\/(ki 2 — k 2 ) + kR] has an extremum at the point where 

k/V(&i 2 -k 2 ) = R/(z + l) = r' sind/r' cosd = tanfl, 



278 



Sound 



§72 



i.e. k = k\ sin #, where 6 is the angle of incidence (see Fig. 35). On changing 
to the path of integration C" which passes through this point at an angle of 
7r/4 to the axis of abscissae, we obtain formula (72.2). 




Fig. 37 



In the case ci < c^ (i.e. h\ > £2), the point k = k\ sin 6 lies between 
&2 and ki if sin 6 > hi\k\ = c\\c^ — sin do, i-e. if 6 > 6q (Fig. 38). In this 
case the contour C" must make a loop round the point kz, and we have, 
besides the ordinary reflected wave (72.2), a wave <f>i" given by the integral 
(72.15) taken around the loop, which we call C" . This is the lateral wave. 
The integral is easily calculated if the point &i sin 9 is not close to hi, i.e. 
if the angle 6 is not close to the internal-reflection angle 0o«t 




Fig. 38 



Near the point k = &2, ft2 is small ; we expand the coefficient of the expo- 
nential in the integrand of (72.15) in powers of fi2. The zero-order term has 
no singularity at k = k%, and its integral round C" is zero. Hence we have 



#>= _ f^L /_^ X p[_(*+/) / , 1+ ;, fj R]dK. 

J MrP2 V 2&77T 
c 



(72.16) 



Expanding the exponent in powers of k — k% and integrating round the loop 



t For an investigation of the lateral wave for all values of d, see L. Brekhovskikh, Zhurnal tekh- 
nicheskoi fiziki 18, 455, 1948. This paper gives also the next term in the expansion of the ordinary 
reflected wave in powers of XjR. We may mention here that, for angles d close to (in the case 
c 1 < c 2 ), the ratio of the correction term to the leading term falls off with distance as (A/i?)i, and 
not as X/R. 



§73 The emission of sound 279 

C", we have after a simple calculation the following expression for the poten- 
tial of the lateral wave : 

<k " = 2*Pifoexp[iftir'cos(e -fl)] 

^ r'a/o^iVtcos^osin^sinS^o-^)]' ' 

In accordance with the previous results, the wave surfaces are the cones 
r' cos(d—9o) = R sin 6o + (z+l) cos do = constant. In a given direction, the 
wave amplitude decreases inversely as the square of the distance r'. We see 
also that this wave disappears in the limit A -> 0. For -> do, the expres- 
sion (72.17) ceases to be valid; in actual fact, the amplitude of the lateral 
wave in this range of 6 decreases with distance as r' -5/4 . 

§73. The emission of sound 

A body oscillating in a fluid causes a periodic compression and rarefaction 
of the fluid near it, and thus produces sound waves. The energy carried 
away by these waves is supplied from the kinetic energy of the body. Thus 
we can speak of the emission of sound by oscillating bodies. f 

In the general case of a body of arbitrary shape oscillating in any manner, 
the problem of the emission of sound waves must be solved as follows. We 
take the velocity potential <f> as the fundamental quantity; it satisfies the wave 
equation 

A^-G/^W/ 3 ' 2 = 0- (73.1) 

At the surface of the body, the normal component of the fluid velocity must 
be equal to the corresponding component of the velocity u of the body: 

8<f>/8n = u n . (73.2) 

At large distances from the body, the wave must become an outgoing spherical 
wave. The solution of equation (73.1) which satisfies these boundary con- 
ditions and the condition at infinity determines the sound wave emitted by 
the body. 

Let us consider the two boundary conditions in more detail. We suppose 
first that the frequency of oscillation of the body is so large that the length 
of the emitted wave is very small compared with the dimension / of the body: 

A < I (73.3) 

In this case we can divide the surface of the body into portions whose dimen- 
sions are so small that they may be approximately regarded as plane, but 
yet are large compared with the wavelength. Then we may suppose that each 



f In what follows we shall always suppose that the velocity u of the oscillating body is small com- 
pared with the velocity of sound. Since u ~ aco, where a is the linear amplitude of the oscillations, 
this means that a <^ A. 

The amplitude of the oscillations is in general supposed small in comparison with the dimensions 
of the body also, since otherwise we do not have potential flow near the body (cf. §9). This con- 
dition is unnecessary only for pure pulsations, when the solution (73.7) used below is really a direct 
deduction from the equation of continuity. 



280 Sound §73 

such portion emits a plane wave, in which the fluid velocity is simply the 
normal component u n of the velocity of that portion of the surface. But the 
mean energy flux in a plane wave is (see §64) cpv 2 , where v is the fluid velocity 
in the wave. Putting v — u n and integrating over the whole surface of the 
body, we reach the result that the mean energy emitted per unit time by 
the body in the form of sound waves, i.e. the total intensity of the emitted 
sound, is 



I = c P j>u n zdf. (73.4) 



It is independent of the frequency of the oscillations (for a given velocity 
amplitude). 

Let us now consider the opposite limiting case, where the length of the 
emitted wave is large compared with the dimension of the body: 

A > I (73.5) 

Then we can neglect the term (llc 2 )d 2 <f>[dt 2 , in the general equation (73.1), 
near the body (at distances small compared with the wavelength). For this 
term is of the order of cd 2 ^/c 2 ~ (f>[X 2 , whereas the second derivatives with 
respect to the co-ordinates are, in this region, of the order of </>// 2 . 

Thus the flow near the body satisfies Laplace's equation, A<£ = 0. This 
is the equation for potential flow of an incompressible fluid. Consequently 
the fluid near the body moves as if it were incompressible. Sound waves 
proper, i.e. compression and rarefaction waves, occur only at large distances 
from the body. 

At distances of the order of the dimension of the body and smaller, the 
required solution of the equation /\<j> = cannot be written in a general form, 
but depends on the actual shape of the oscillating body. At distances large 
compared with /, however (though still small compared with A, so that the 
equation /\<j> = remains valid), we can find a general form of the solution 
by using the fact that <f> must decrease with increasing distance. We have 
already discussed such solutions of Laplace's equation in §11. As there, 
we write the general form of the solution as 

cj> = -(a/r)+A.grad(l/r), (73.6) 

where r is the distance from an origin anywhere inside the body. Here, of 
course, the distances involved must be large compared with the dimension 
of the body, since we cannot otherwise restrict ourselves to the terms in <j> 
which decrease least rapidly as r increases. We have included both terms in 
(73.6), although it must be borne in mind that the first term is sometimes 
absent (see below). 

Let us ascertain in what cases this term — a/r is non-zero. We found 
in §11 that a potential —a\r results in a non-zero value Airpa of the mass flux 
through a surface surrounding the body. In an incompressible fluid, how- 
ever such a mass flux can occur only if the total volume of fluid enclosed within 



§73 The emission of sound 281 

the surface changes. In other words, there must be a change in the volume 
of the body, as a result of which the fluid is either expelled from or "sucked 
into" the volume of space concerned. Thus the first term in (73.6) appears 
in cases where the emitting body undergoes pulsations during which its 
volume changes. 

Let us suppose that this is so, and determine the total intensity of the 
emitted sound. The volume Aira of the fluid which flows through the closed 
surface must, by the foregoing argument, be equal to the change per unit time 
in the volume V of the body, i.e. to the derivative dVjdt (the volume V 
being a given function of the time) : 4rra = V. Thus, at distances r such 
that / <^ r <^ A, the motion of the fluid is given by the function <f> = — V(t)\$Trr. 
At distances r > A, however (i.e. in the "wave region"), <f> must represent an 
outgoing spherical wave, i.e. must be of the form 

ftt-rlc) 

4> = -- l -l. (73.7) 

r 

Hence we conclude at once that the emitted wave has, at all distances large 
compared with /, the form 

V(t-rlc) 
<f> = ~ K ' \ (73.8) 

477T 

which is obtained by replacing the argument t of (tV) by t—rjc. 

The velocity v = grad cf> is directed at every point along the radius 
vector, and its magnitude is v = d<f>jdr. In differentiating (73.8) for distances 
r > A, only the derivative of the numerator need be taken, since differentiation 
of the denominator would give a term of higher order in 1/r, which we neglect. 
Since W{t-r\c)]dr = -(l/c)F(*-r/c), we obtain 

v = V(t-r/c)n/4iTcr, (73.9) 

where n is a unit vector in the direction of r. 

The intensity of the sound is given by the square of the velocity, and is 
here independent of the direction of emission, i.e. the emission is isotropic. 
The mean value of the total energy emitted per unit time is 

I = pc jtfdf = ( P /16c7T 2 ) j (7 2 /r2)d/, 

where the integration is taken over a closed surface surrounding the origin. 
Taking this surface to be a sphere of radius r, and noticing that the integrand 
depends only on the distance from the origin, we have finally 

/ = p V 2 /4ttc. (73.10) 

This is the total intensity of the emitted sound. We see that it is given by 
the squared second time derivative of the volume of the body. 



282 Sound §73 

If the body executes harmonic pulsations of frequency to, the second time 
derivative of the volume is proportional to the frequency and velocity 
amplitude of the oscillations, and its mean square is proportional to the 
square of the frequency for a given velocity amplitude of points on the surface 
of the body. For a given amplitude of the oscillations, however, the velocity 
amplitude is itself proportional to the frequency, so that the intensity of 
emission is proportional to co 4 . 

Let us now consider the emission of sound by a body oscillating without 
change of volume. Only the second term then remains in (73.6) ; we write it 
<f> = div[A(t)[r]. As in the preceding case, we conclude that the general 
form of the solution at all distances r p I is <f> — div[A(£— r/c)jr]. That 
this expression is in fact a solution of the wave equation is seen immediately, 
since the function A(t—r/c)/r is a solution, and therefore so are its derivatives 
with respect to the co-ordinates. Again differentiating only the numerator, 
we obtain (for distances r > A) 

<f> = -A(t-rjc)-nlcr. (73.11) 

To calculate the velocity v = grad</>, we need again differentiate only A. 
Hence we have, by the familiar rules of vector analysis for differentiation 
with respect to a scalar argument, 

A(/-r/c)-n 



v = — 



^grad(.-I), 



cr 

and, substituting gTad(t-rjc) = -(l/<:)gradr = -n/c, we have finally 

v = n(n-A)/c2r. (73.12) 

The intensity is now proportional to the squared cosine of the angle between 
the direction of emission (i.e. the direction of n) and the vector A; this is 
called dipole emission. The total emission is given by the integral 



c 3 J r 

We again take the surface of integration to be a sphere of radius r, and use 
spherical co-ordinates with the polar axis in the direction of the vector A. 
A simple integration gives finally for the total emission per unit time 

/=^A2. (73.13) 

The components of the vector A are linear functions of the components of 
the velocity u of the body (see §11). Thus the intensity is here a quadratic 
function of the second time derivatives of the velocity components. 

If the body executes harmonic oscillations of frequency to, we conclude 
(reasoning as in the previous case) that the intensity is proportional to co 4 



§73 The emission of sound 283 

for a given value of the velocity amplitude. For a given linear amplitude of 
the oscillations of the body, the velocity amplitude is proportional to the 
frequency, and therefore the intensity is proportional to w 6 . 

In an entirely similar manner we can solve the problem of the emission of 
cylindrical sound waves by a cylinder of any cross-section pulsating or 
oscillating perpendicularly to its axis. We shall give here the corresponding 
formulae, with a view to later applications. 

Let us first consider small pulsations of a cylinder, and let S = S(t) be 
its (variable) cross-sectional area. At distances r from the axis of the cylinder 
such that / < r 4. A, where / is the transverse dimension of the cylinder, 
we have similarly to (73.8) 

<f> = [S(t)l27T]\ogfr y (73.14) 

where /(i) is a function of time, and the coefficient of log/r is chosen so as to 
obtain the correct value for the mass flux through a coaxial cylindrical surface. 
In accordance with the formula for the potential of an outgoing cylindrical 
wave (the first term of formula (70.2)), we now conclude that at all distances 
r > / the potential is given by 



t-rlc 



c_ r ^t)df 

^ Jo V[c 2 (t-t'f-r2]' K ' ' 

As r -> the leading term of this expression is the same as (73.14), and 
the function f(t) in the latter equation is automatically determined (we 
suppose that the derivative S(t) tends sufficiently rapidly to zero as t -> — oo). 
For very large values of r, on the other hand (the "wave region"), the values 
oi t—t' ~ rjc are the most important in the integral (73.15). We can there- 
fore put, in the denominator of the integrand, 



(t-tf-i*l<* « (2r/c)(*-*'-r/c), 



obtaining 



c *"f S(t')dt' 
£= Li . (73.16) 

Finally, the velocity v = d<f>/dr. To effect the differentiation, it is con- 
venient to substitute t — t' — rjc = £: 

6= -- I C 1 fo-^-fl df- 

the limits of integration are then independent of r. The factor l/- v /r in 
front of the integral need not be differentiated, since this would give a term 



284 Sound §73 

of higher order in 1/r. Differentiating under the integral sign and then 
returning to the variable t', we obtain 

t-r/c 

S(t')dt' 

v = 



r S(f)dt' 

— . (73.17) 



WW i Vi<t-t')-r] 

The intensity is given by the product iTrrpcv 2 . It should be noticed that here, 
unlike what happens for the spherical case, the intensity at any instant is 
determined by the behaviour of the function S(t) at all times from — oo 
to t — rjc. 

Finally, for translatory oscillations of an infinite cylinder in a direction 
perpendicular to its axis, the potential at distances r such that / <^ r <^ A 
has the form 

<f> = div(A log fr), (73.18) 

where A(t) is determined by solving Laplace's equation for the flow of an 
incompressible fluid past a cylinder. Hence we again conclude that, at all 
distances r > /, 



t-r/c 



r Alt' )dr 
<f> = -div — . (73.19) 

In conclusion, we must make the following remark. We have here entirely 
neglected the effect of the viscosity of the fluid, and accordingly have sup- 
posed that there is potential flow in the emitted wave. In reality, however, we 
do not have potential flow in a fluid layer of thickness ~ \/(vJco) round the 
oscillating body (see §24). Hence, if the above formulae are to be applicable, 
it is necessary that the thickness of this layer should be small in comparison 
with the dimension / of the body: 

VWco) < /• (73.20) 

This condition may not hold for small frequencies or small dimensions of 
the body. 

PROBLEMS 

Problem 1. Determine the total intensity of sound emitted by a sphere executing small 
(harmonic) translatory oscillations of frequency on, the wavelength being comparable in 
magnitude with the radius R of the sphere. 

Solution. We write the velocity of the sphere in the form u = u e~ i(Ot ; then <f> depends 
on the time through a factor e~ ib)t also, and satisfies the equation A<f>+k 2 <f> — 0, where k = tojc. 
We seek a solution in the form <j> = u • grad/(r), the origin being taken at the instantaneous 
position of the centre of the sphere. For / we obtain the equation u • grad( Af+k 2 f) = 0, 
whence Af+k 2 f = constant. Apart from an unimportant additive constant, we therefore 
have / = Ae ikT lr. The constant A is determined from the condition 8<f>/dr = u r for r = R, 
and the result is 

/R\3 ikr-1 

d> = wre ik(r - R) i — . 

r \ r I 2-2ikR-k 2 R 2 



§73 The emission of sound 285 

Thus we have dipole emission. At fairly large distances from the sphere, we can neglect 
unity in comparison with ikr, and <f> takes the form (73.11), the vector A being 

ia> 
A = -ue ik(r - R) R 3 - 



2-2ikR-k*R 2 



Noticing that (re A) 2 = i|A| 2 , we obtain for the total emission, by (73.13), 

2tt P , , i? 6 eo 4 

= — N 



3c3' ' 4 + (coR/cy 

For toRfc <^ 1, this expression becomes / = irpR 6 \u \ 2 co 4 l6c 6 , a result which could also be 
obtained by directly substituting in (73.13) the expression A = £R s u from §11, Problem 1. 
For o>i?/c^> 1 we have 1 = 27rpci? 2 |u | 2 /3, corresponding to formula (73.4). 

The drag force acting on the sphere is obtained by integrating over the surface of the 
sphere the component of the pressure forces (p' — — p(^')r=fe) in the direction of u, and is 

4tt -kW3 + i(2+k 2 R*) 
F = — pcoR 3 u ; 

see the end of §24 concerning the meaning of a complex drag force. 

Problem 2. The same as Problem 1, but for the case where the radius R of the sphere is 
comparable in magnitude with V( •'/<")» whilst A^> R. 

Solution. If the dimension of the body is small compared with \/( ''/<«»)> then the emitted 
wave must be investigated not from the equation A ^ = 0, but from the equation of motion 
of an incompressible viscous fluid. The appropriate solution of this equation for a sphere is 
given by formulae (1) and (2) in §24, Problem 5. At great distances the first term in (1), 
which diminishes exponentially with r, may be omitted. The second term gives the velocity 
v = — i(u # grad)grad(l/r). Comparison with (73.6) shows that 

A = -bu = lR3[l-3l(i-l)K-3/2iK2]\i, 

where #c = Ry/(u>l2v), i.e. A differs from the corresponding expression for an ideal fluid 
by the factor in brackets. The result is 

ttdR* / 3 9 9 9 \, , 

I = J— a>M 1+- + + + uo p. 

For k ^> 1 this becomes the formula given in Problem 1 , while for ic^lwe obtain 

/ = 3tt P R 2 vV\uo\ 2 I2(?, 

i.e. the emission is proportional to the second, and not the fourth, power of the frequency. 

Problem 3. Determine the intensity of sound emitted by a sphere executing small (har- 
monic) pulsations of arbitrary frequency. 

Solution. We seek a solution of the form <f> = {aujr)e ik ^ r ~ Ki , R being the equilibrium 
radius of the sphere, and determine the constant a from the condition [d4>Jdr] r =R = u 
= u e~ i0)t (where u is the radial velocity of points on the surface of the sphere): 

a = R 2 l{ikR-\). 

The intensity is I = 27rpc|u | 2 # ! -R 4 /(l +k 2 R i ). For kR < 1, / = 2ir P a) 2 R i \u \ 2 /c, in accor- 
dance with (73.10), while for kR ^> 1, / = 2npcR 2 \u \ 2 , in accordance with (73.4). 

Problem 4. Determine the nature of the wave emitted by a sphere (of radius R) executing 
small pulsations, when the radial velocity of points on the surface is an arbitrary function 
u(t) of the time. 



286 Sound §73 

Solution. We seek a solution in the form <f> =f(t')Jr, where t' = t—{r—R)jc, and deter- 
mine / from the boundary condition d<f>j8r = u(t) for r = R. This gives the equation 
dfJdt+cf(t)JR = — Rcu(t). Solving this linear equation and replacing t by t' in the solution, 
we obtain 

„ t 

cR r 

<f,(r, t') = e-<*/R u{t)(*»/R&t. (1) 

—oo 

If the oscillations of the sphere cease at some instant, say t — (i.e. u(j) = for r > 0), 
then the potential at a distance r from the centre will be of the form <f> = constant X e~ e ' ls 
after the instant t = (r—R)Jc, i.e. it will diminish exponentially. 

Let T be the time during which the velocity u(t) changes appreciably. If T ^> Rjc, i.e. if 
the wavelength of the emitted waves A ~ cT^> R, then we can take the slowly varying factor 
«(t) outside the integral in (1), replacing it by u(t'). For distances r^> R, we then obtain 
<f> = -~(R 2 Jr)u(t—rjc), in accordance with formula (73.8). If, on the other hand, T <^.RJc, 
we obtain in a similar manner 

cR C 
$ = w ( T )d T , v = 8^1 8r = {Rlr)u{t'), 

—oo 
in accordance with formula (73.4). 

Problem 5. Determine the motion of an ideal compressible fluid when a sphere of radius 
R executes in it an arbitrary translatory motion, with velocity small compared with that of 
sound. 

Solution. We seek a solution in the form 

$ = div[f(*>], (1) 

where r is the distance from the origin, taken at the position of the centre of the sphere at 
the time t' = t—(r—R)/c; since the velocity u of the sphere is small compared with the 
velocity of sound, the movement of the origin may be neglected. The fluid velocity is 

„ aa 3(f.n)n-f t 3(f'.n)n-f' , (f".n)n 
v = grad^ = + + , (2) 

where n is a unit vector in the direction of r, and the prime denotes differentiation with 
respect to the argument of f. The boundary condition is v r = u • n for r — R, whence 
f"(t)+(2clR)f'(t)+(2c*IR 2 )f(t) = Rc 2 u(t). Solving this equation by variation of the para- 
meters, we obtain for the function f(t) the general expression 

* ( \ 

f(t) = cR*e-*i* fu(T)sin-^— IV/-Rc1t. (3) 

—00 

In substituting in (1), we must replace t by t'. The lower limit is taken as — oo so thatf 
shall be zero for t = — oo. 

Problem 6. A sphere of radius i? begins at time t = to move with constant velocity u . 
Determine the sound intensity emitted at the instant when the motion begins. 

Solution. Putting in formula (3) of Problem 5 u(t) = f or t < and u(t) = u for 
t > 0, and substituting in formula (2) (retaining only the last term, which decreases least 
rapidly with r), we find the fluid velocity far from the sphere : 



v = — n(n«uo) 



\/2R l ct' \ 
er<*i R sin I M, 



§73 The emission of sound 287 

where t' > 0. The total intensity diminishes with time according to 

I = (87rl3)cpR 2 uo 2 e-^' / Rsm z (ct'IR-i7r). 

The total amount of energy emitted is iTrpi? 3 u 2 . 

Problem 7. Determine the intensity of sound emitted by an infinite cylinder, of radius R, 
executing harmonic pulsations of wavelength A ^> R. 

Solution. According to formula (73.14), we find first of all that, at distances r <^ A 
(in Problems 7 and 8 r is the distance from the axis of the cylinder), the potential is 
<f> = Ru log kr, where u = u a e~ im is the velocity of points on the surface of the cylinder. 
From a comparison with formulae (70.7) and (70.8), we now find that at large distances the 
potential is of the form 4> = —Ru^{iirl2kr)e ikr . The velocity is therefore 

v = Ru^/(7rk/2ir)ne ikr , 

where n is a unit vector perpendicular to the axis of the cylinder, and the intensity per unit 
length of the cylinder is / = $n 2 po>R 2 u 2 . 

Problem 8. Determine the intensity of sound emitted by a cylinder executing harmonic 
translatory oscillations in a direction perpendicular to its axis. 

Solution. At distances r <^ A we have <j> — — div(i? 2 u log kr) ; cf. formula (73.18) and 
§10, Problem 3. Hence we conclude that at large distances 

<j> = ^V^^divC^^u/Vr) = -R 2 u.n^/(7Tkl2ir)eMr t 

whence the velocity is v = — kR 2 \/(inkl2r)n(vfn)e ikr . The intensity is proportional to the 
squared cosine of the angle between the directions of oscillation and emission. The total 
intensity is J = (7r 2 /4c 2 )pa) 3 i2 4 |u | 2 . 

Problem 9. Determine the intensity of sound emitted by a plane surface whose temperature 
varies periodically with frequency o» <^ e 2 /x» where x is the thermometric conductivity of the 
fluid. 

Solution. Let the variable part of the temperature of the surface be T / 9 e~ i<ot . These 
temperature oscillations cause a damped thermal wave in the fluid (52.17): 

T' = T , oe- i<ot e~ a - i) ^ < - b>/2 x )x i 

and the fluid density therefore oscillates also: />' = (dp\dT) v T = —pPT', where P is the 
coefficient of thermal expansion. This, in turn, results in the occurrence of a motion deter- 
mined by the equation of continuity: p dv/dx = —dp'jdt = —imp^T'. At the solid surface 
the velocity v x = v = 0, and far from the surface it tends to the limit 

00 ^ 

v = -imp f T'dx = -^Lp^ajfiT'oer** 
o 

This value is reached at distances , ^ J V(xl <0 )> which are small compared with c/to, and we 
thus have a boundary condition on the resulting sound wave. Hence we find the intensity 
per unit area of the surface to be / = icp^ 2 wx\T / e \ 2 . 

Problem 10. A point source emitting a spherical wave is at a distance / from a solid wall 
which totally reflects sound and bounds a half-space occupied by fluid. Determine the ratio 
of the total intensity of sound emitted by the source to that which would be found in an infinite 
medium, and the dependence of the intensity on direction for large distances from the source. 

Solution. The sum of the direct and reflected waves is given by a solution of the wave 
equation such that the normal velocity component v n — 3<f>/dn is zero at the wall. Such a 
solution is 

oikr pikr' 



- + —)-<» 



288 Sound §74 

(we omit the constant factor, for brevity), where r is the distance from the source O (Fig. 
39), and r' is the distance from a point O' which is the image of O in the wall. At large dis- 
tances from the source we have r' 7H r— 21 cos 0, so that 

e i(kr— <ot) 
(f) = (l-f e -2ta?cos<?). 

r 

The dependence of the intensity on direction is given by a factor cos 2 (&/ cos 0). 

To determine the total intensity, we integrate the energy flux q = p'v = — p<£ grad 4 
(see (64.4)) over the surface of a sphere of arbitrarily small radius, centred at O. This gives 
2irpk<*>{\ +[l/2&/] sin 2kl). In an infinite medium, on the other hand, we should have simply a 
spherical wave <p = e i(*r-«t)/ r> w ith a total energy flux 2irpka>. Thus the required ratio of 
intensities is 1 +(l/2kl) sin 2kl. 






3 6 e - / - 



^-/--^. 



Fig. 39 



Problem 11. The same as Problem 10, but for a fluid bounded by a free surface. 

Solution. At the free surface the condition p' = — p4> = must hold; in a monochro- 
matic wave this is equivalent to ^ = 0. The corresponding solution of the wave equation is 

oikr t>ikf 

\g—io>t 



(gtKr e i/cr \ 



At large distances from the source, the intensity is given by a factor sin 2 (kl cos 8). The re- 
quired ratio of intensities is 1 —(\j2kl) sin 2kl. 

§74. The reciprocity principle 

In deriving the equations of a sound wave in §63, it was assumed that the 
wave is propagated in a homogeneous medium. In particular, the density 
po of the medium and the velocity of sound in it, c, were regarded as constants. 
In order to obtain some general relations applicable for an arbitrary inhomo- 
geneous medium, we shall first derive the equation for the propagation of 
sound in such a medium. 

We write the equation of continuity in the form dpjdt+pdivv = 0. 
Since the propagation of sound is adiabatic, we have 



dt \8plsdt c* dt c*\dt * F ] 



and the equation of continuity becomes #p/d* + v«grad/> + pc 2 div v = 0. 

As usual, we put p — po + p', where po is now a given function of the 

co-ordinates. In the equation p = po +p', however, we must put as before 



§74 The reciprocity principle 289 

po = constant, since the pressure must be constant throughout a medium in 
equilibrium (in the absence of an external field, of course). Thus we have to 
within second-order quantities dp'/dt+poc 2 div v = 0. 

This equation is the same in form as equation (63.5), but the coefficient 
poc 2 is a function of the co-ordinates. As in §63, we obtain Euler's equation 
in the form dvjdt = -(1/po) gradp'. Eliminating v, and omitting the 
suffix in po, we finally obtain the equation of propagation of sound in an 
inhomogeneous medium: 

&&L-1-2L-0. (74,) 

p pc l ct* 

If the wave is monochromatic, with frequency w, we have p' = — co 2 />', 
so that 

d . v grad/ + ^, = o (742) 

P P& 

Let us consider a sound wave emitted by a pulsating source of small 
dimension; we have seen in §73 that the emission is isotropic. We denote by 
A the point where the source is, and by pjfi) the pressure p' at a point B 
in the emitted wave.f If the same source is placed at B, it produces at A 
a pressure which we denote by Pb(A). We shall derive the relation between 
p A (B) and p B (A). 

To do so, we use equation (74.2), applying it first to the sound from a 
source at A and then to the sound from a source at B: 

eradp' A (o 2 , n ,. gradp' B co 2 

d iv« L± + p' A = 0, div-2 — + —Tp's = 0. 

P pc 2 p pr 

We multiply the first equation by p '# and the second by p' a and subtract. 
The result is 

, gra&pA . ,. gradp's 
p' B div p a div 

P P 

_ di / P'b g™&P'A P'a gradp'e \ = Q 

\ P P I 

We integrate this equation over the volume between an infinitely distant 
closed surface C and two small spheres Ca and Cb which enclose the points 
A and B respectively. The volume integral can be transformed into three 
surface integrals, and the integral over C is zero, since the sound field vanishes 
at infinity. Thus we obtain 

r ^ *«£*__, VWb\ m _ i (74 . 3) 



t The dimension of the source must be small compared with the distance between A and B and 
with the wavelength. 



290 Sound §74 

Inside the small sphere C A , the pressure p' A in the wave from a source 
at A falls off rapidly with the distance from A, and the gradient gradp' A 
is therefore large. The pressure p' B due to a source at B is a slowly varying 
function of the co-ordinates in the region near the point A, which is at a 
considerable distance from B, so that the gradient gradp' B is relatively small. 
When the radius of the sphere C A is sufficiently small, therefore, we can 
neglect the integral 

j(p'A/p)gradp' B -d£ 

over C A in comparison with 

J(P'b/p) gradp' A .df, 

and in the latter the almost constant quantity p' B can be taken outside the 
integral and replaced by its value at the point A. Similar arguments hold for 
the integrals over the sphere C B , and as a result we obtain from (74.3) the 
relation 



But (l[p)gradp' = -dvfdt, and this equation can therefore be rewritten 

a 



c A 
The integral 



p' B (A)—j>v A .df = p' A (B)j-j>v B .df. 



<fv^.df 



over C A is the volume of fluid flowing per unit time through the surface of 
the sphere C A , i.e. it is the rate of change of the volume of the pulsating 
source of sound. Since the sources at A and B are identical, it is clear that 

fv^.df=fv B .df, 

and consequently 

p' A {B) = p' B {A). (74.4) 

This equation constitutes the reciprocity principle : the pressure at B due 
to a source at A is equal to the pressure at A due to a similar source at B. 
It should be emphasised that this result holds, in particular, for the case 
where the medium is composed of several different regions, each of which 



§75 Propagation of sound in a tube 291 

is homogeneous. When sound is propagated in such a medium, it is reflected 
and refracted at the surfaces separating the various regions. Thus the reci- 
procity principle is valid also in cases where the wave undergoes reflection 
and refraction on its path from A to B. 

PROBLEM 

Derive the reciprocity principle for dipole emission of sound by a source which oscillates 
without change of volume. 



Solution. In this case the integral 



^V^-df 



over C A is zero identically, and the next approximation must be taken in calculating the 
integrals in (74.3). To do so, we write, as far as the first-order terms, 

P'b = p'^ + T-ffradp's, 

where r is the radius vector from A. In the integral 

, grad^ , gr ade's \ Ar (1) 



f^*^-,j£±t>yn 



the two terms are now of the same order of magnitude. Substituting here for p' B the above 
expansion, and using the fact that the integral 



j>(ll P )gradp' A >d£ 

over Ca is now zero, we obtain 

<b (r- grade's) p a j 



Next, we take the almost constant quantity gradp' B = -pv B outside the integral, replacing 
it by its value at A : 

C A P 
where pa is the density of the medium at the point A To calculate this integral, we notice 
that near a source the fluid can be supposed incompressible (see §73), and hence we can write 
for the pressure inside the small sphere Ca, by (11.1), P'a = -p+ = pA-r/r 3 . In a mono- 
chromatic wave v = — icov, A = —itoA; introducing also the unit vector tla in the direction 
of the vector A for a source at A, we find that the integral (1) is proportional to p A v B (A) • n^. 
Similarly, the integral over the sphere C B is proportional to — pbva(B) • n Bl with the same 
factor of proportionality. Equating the sum to zero, we find the required relation 

PA^B(A)'n A = pBVA{B)-n B , 

which expresses the reciprocity principle for dipole emission of sound. 

§75. Propagation of sound in a tube 

Let us now consider the propagation of a sound wave in a long narrow tube. 
By a "narrow" tube we mean one whose width is small compared with the 



292 Sound 



§75 



wavelength. The cross-section of the tube may vary along its length in both 
shape and area. It is important, however, that this variation should occur 
fairly slowly: the cross-sectional area S must vary only slightly over distances 
of the order of the width of the tube. 

Under these conditions we can suppose that all quantities (velocity, 
density, etc.) are constant over any transverse cross-section of the tube. The 
direction of propagation of the wave can be supposed to coincide with that of 
the axis of the tube at all points. The equation for the propagation of such 
a wave is most conveniently derived by a method similar to that used in §13 
in deriving the equation for the propagation of gravity waves in channels. 

In unit time a mass Spv of fluid passes through a cross-section of the tube. 
Hence the mass of fluid in the volume between two transverse cross-sections 
at a distance dx apart decreases in unit time by 

{S P v) x+&x -{Spv) x = [d(S P v)/8x]dx, 
the co-ordinate x being measured along the axis of the tube. Since the volume 
between the two cross-sections remains constant, the decrease must be due 
only to the change in density of the fluid. The change in density per unit time 
is dp/dt, and the corresponding decrease in the mass of fluid in the volume 
S dx between the two cross-sections is -S{8pj8t)dx. Equating the two 
expressions, we obtain 

S8p/8t = -8(Spv)ldx, (75.1) 

which is the "equation of continuity" for flow in a pipe. 

Next, we write down Euler's equation, omitting the term quadratic in the 
velocity: 

dv/dt= -(l/p)8p/8x. (75.2) 

We differentiate (75.1) with respect to time, regarding p on the right-hand 
side as independent of time, since the differentiation of p gives a term which 
involves v dp/dt = v dpjdt and is therefore of the second order of smallness. 
Thus S 8 2 p\8t 2 = - 8{Sp8vj8t)J8x. Here we substitute the expression (75.2) 
for 8v/8t, and express the derivative of the density on the left-hand side in 
terms of the derivative of the pressure by p = (8p[8p)p = p/c 2 . 

The result is the following equation for the propagation of sound in a 
tube: 

1 8 I dp\ 1 8 2 p 

In a monochromatic wave p depends on time through a factor e _iw ', and 
(75.3) becomes 

1 8 l 8p\ 

-sT x \ S £) +k2p = (i - < 75 ' 4 > 

where k — co/c is the wave number.f 



f Here, and in the Problems, p denotes the variable part of the pressure, which we have previ- 
ously denoted by p'. 



§75 Propagation of sound in a tube 293 

Finally, let us consider the problem of the emission of sound from the 
open end of a tube. The pressure difference between the gas in the end of 
the tube and that in the space surrounding the tube is small compared with the 
pressure differences within the tube. Hence the boundary condition at the 
open end of the tube is, with sufficient accuracy, that the pressure p should 
vanish. The gas velocity v at the end of the tube is not zero ; let its value be 
vq. The product Svq is the volume of gas leaving the tube per unit time. 

We can now regard the open end of the tube as a "source" of gas of strength 
Svq. The problem of the emission from a tube thus becomes equivalent to 
that of the emission by a pulsating body, which is solved by formula (73.10). 
In place of the time derivative Voi the volume of the body we must now put 
Svo. Thus the total intensity of the sound emitted is 

/ = pS^IAttc. (75.5) 

PROBLEMS 

Problem 1. Determine the transmission coefficient for sound passing from a tube of cross- 
section S x into one of cross-section S 2 . 

Solution. In the first tube we have two waves, the incident wave p x — a 1 e iikx ~ a>t) and 
the reflected wave p x ' = a{ ' e -M*+<»t) . In the second tube we have the transmitted wave 
p 2 = a 2 e^ kx - m \ At the point where the tubes join (x = 0), the pressures must be equal, 
and so must the volumes Sv of gas passing from one tube to the other per unit time. These 
conditions give aj+fli" = <h, S 1 (a 1 -a 1 ') = S 2 a 2 , whence a 2 = 2a 1 5 1 /( ( S , 1 + 4 S 2 ). The ratio D 
of the energy flux in the transmitted wave to that in the incident wave is 

D = S 2 pc\^\zjS iP c\^\z = S 2 |^/£iM2 

or 

4SlS2 , /S2-Si\ 2 



D = 



\s*+sj 



Problem 2. Determine the amount of energy emitted from the open end of a cylindrical 
tube. 

Solution. In the boundary condition p = at the open end of the tube, we can approxi- 
mately neglect the emitted wave (we shall see that the intensity emitted from the end of the 
tube is small). Then we have the condition p x = —p x \ where p t andpi are the pressures in 
the incident wave and in the wave reflected back into the tube ; for the velocities we have 
correspondingly v t = Vy, so that the total velocity at the end of the tube is v = Vx+v^ = 2v t . 
The energy flux in the incident wave is cSpv} = icSpv^. Using (75.5), we obtain for the 
ratio of the emitted energy to the energy flux in the incident wave D = Su) 2 /nc 2 . For a 
tube of circular cross-section (radius R) we have D = i? a w 2 /c 2 . Since, by hypothesis R<€clo> 
it follows that D<1. » \ / , 

Problem 3. One of the ends of a cylindrical pipe is covered by a membrane which executes 
a given oscillation and emits sound ; the other end is open. Determine the way in which sound 
is emitted from the tube. 

Solution. In the general solution 

p = (a e ikx + J) e -i1cxy-iut 

we determine the constants a and b from the conditions v = u = M e _i& ", the given velocity 
of the membrane, at the closed end (x = 0), and p = at the open end (x = /). These give 



294 Sound §76 

ae lke +be~ m = 0, a—b = cpu . Determining a and b, we find the gas velocity at the open 
end of the tube to be v — w/cos kl. If the tube were absent, the intensity of the sound emitted 
by the oscillating membrane would be given by the mean square <S 2 Itt| 2 = 5 2 w 2 |m| 2 , according 
to formula (73.10) with Su in place of V; S is the cross-sectional area of the membrane. 
The emission from the end of the tube is proportional to S 2 \v \ 2 a) 2 . Defining the "amplifi- 
cation coefficient" of the pipe as A = S a \v \*/S a \u[*, we obtain A — 1/cos 2 kl. This becomes 
infinite for frequencies of oscillation of the membrane equal to the characteristic frequencies 
of the tube {resonance) ; in reality, of course, it remains finite because of effects which we have 
neglected (such as friction due to the emission of sound). 

Problem 4. The same as Problem 3, but for a conical tube, with the membrane covering 
the smaller end. 

Solution. The cross-section of the tube is S = SqX 2 ; let the values of the co-ordinate x 
which correspond to the smaller and larger ends be x lt x 2 , so that the length of the tube is 
/ = x 2 —Xi. The general solution of equation (75.4) is p = (l/x)(ae ilcx +be~ ikx )e~ im ; a and b 
are determined from the conditions v = u for x = Xi and p = for * = x 2 . The amplifica- 
tion coefficient is found to be 

SoX2 2 \v2\ 2 k 2 Xi 2 

A 



Sox± 2 1 u | 2 (sin kl + kxi cos kl) 2 

Problem 5. The same as Problem 3, but for a tube whose cross-section varies exponen- 
tially along its length: S = Soe xx . 

Solution. Equation (75.4) becomes d 2 pl8x 2 +adpjdx+k 2 p = 0, whence 
p = e -\*x( ae imx + b e -imxy-iuit t 

with m = -\/{k 2 —\a. 2 ). Determining a and b from the conditions v = u for x = and p — 
for x = /, we find the amplification coefficient 

Soe al \vq\ 2 e al 



So\u\ 2 [£( a / m ) s i n m ^+ cos m ^] 2 

for k > $a and 



[|-(a/m') sinh m'l+ cosh m'Vf V KZ h 



for k < ^a. 



§76. Scattering of sound 

If there is some body in the path of propagation of a sound wave, then the 
sound is scattered: besides the incident wave there appear other (scattered) 
waves, which are propagated in all directions from the scattering body. The 
scattering of a sound wave occurs simply on account of the presence of the 
body in its path. In addition, the incident wave causes the body itself to move, 
and this in turn brings about additional emission of sound by the body, i.e. 
further scattering. If, however, the density of the body is large compared 
with that of the medium in which the sound is propagated, and its compres- 
sibility is small, then the scattering due to the motion of the body forms only a 
small correction to the main scattering caused by the mere presence of the 
body. In what follows we shall neglect this correction, and therefore suppose 
the scattering body immovable. 



§76 Scattering of sound 295 

We assume that the wavelength A of the sound is large compared with the 
dimension / of the body; to calculate the properties of the scattered wave, 
we can then use formulae (73.8) and (73.11).j- In doing so, we regard the 
scattered wave as being emitted by the body; the only difference is that, 
instead of a motion of the body in the fluid, we now have a motion of the fluid 
relative to the body. The two problems are clearly equivalent. 

For the potential of the emitted wave we have obtained the expression 
<f> = — VjAttt — A* rjcr 2 . In this formula V was the volume of the body. 
In the present case, however, the volume of the body itself remains unchanged, 
and V must be taken not as the rate of change of the volume of the body, but 
as the volume of fluid which would enter, per unit time, the volume Vq 
occupied by the body if the body were absent. For, in the presence of the 
body, this volume of fluid does not penetrate into Vq, which is equivalent to 
the emission of the same volume of fluid from Vq. The coefficient of 1/477T 
in the first term of <f> must, as we have seen in §73, be just the volume of fluid 
emitted from the origin per unit time. This volume is easily found. The 
change per unit time in the mass of fluid in a volume equal to that of the body 
is Vop, where p gives the rate of change of the fluid density in the incident 
sound wave (since the wavelength is large compared with the dimension 
of the body, the density p may be supposed constant over distances of the 
order of this dimension; hence we can write the rate of change of the mass of 
fluid in Vq as Vop simply, where p is the same throughout the volume Vq). 
The change in volume corresponding to a mass change Vop is evidently 
Vopfp. Thus V in the expression for (f> must be replaced by Vop/ p. In an 
incident plane wave, the variable part p of the density is related to the velocity 
by p = pv/c; hence p = p' = pvfc, and we can replace Vopfp by Vqvjc. 

When the body moves in the fluid, the vector A is determined by formulae 
(11.5), (11.6): AnpAi = miicUk+ pVoUi. We must now replace the velocity 
u of the body by the reversed velocity v of the fluid in the incident wave which 
it would have at the position of the body if the latter were absent. Thus 

At = —m ilc V]clATTp—VoViJ^TT. (76.1) 

We finally obtain for the potential of the scattered wave 

0sc= -Vov\\Trcr-k*T\cr\ (76.2) 

the vector A being given by formula (76.1). Hence we have for the velocity 
distribution in the scattered wave 

v sc = Vq vnl47rrc 2 + n(n • A)frc 2 (76.3) 

(see §73), n being a unit vector in the direction of scattering. 

The mean amount of energy scattered per unit time into a given solid angle 
element do is given by the energy flux, which is cpv S c 2 r 2 do. The total scat- 
tered intensity 7 S c is obtained by integrating this expression over all directions. 



f At the same time, the dimension of the body must be large in comparison with the displacement 
amplitude of fluid particles in the wave, since otherwise the fluid is not in general in potential flow. 



296 Sound §76 

The integration of twice the product of the two terms in (76.3) gives zero, 
since this product is proportional to the cosine of the angle between the 
direction of scattering and the direction of propagation of the incident wave, 
and there remains (cf. (73.10) and (73.13)) 

The scattering is generally characterised by what is called the effective 
cross-section do-, which is the ratio of the (time) average energy scattered into 
a given solid-angle element to the mean energy flux density in the incident 
wave. The total effective cross-section a is the integral of da over all directions 
of scattering, i.e. it is the ratio of the total scattered intensity to the incident 
energy flux density, and evidently has the dimensions of area. 

The mean energy flux density in the incident wave is cp\ 2 . Hence the 
differential effective scattering cross-section is (cpv S c 2 /c/>v 2 )r 2 do, i.e. 

do = (v^2/v2)r2do. (76.5) 

The total effective cross-section is 

V 2 V 2 " 4tt A* 

a = t= + — •=■. (76.6) 

477-c 4 v2 3c 4 V2 v ' 

For a monochromatic incident wave, the mean square second time derivative 
of the velocity is proportional to the fourth power of the frequency. Thus the 
effective cross-section for the scattering of sound by a body which is small 
compared with the wavelength is proportional to oA. 

Finally, let us briefly discuss the opposite limiting case, where the wave- 
length of the scattered sound is small compared with the dimension of the 
body. In this case all the scattering, except for the scattering through 
very small angles, amounts to simple reflection from the surface of the body. 
The corresponding part of the total effective scattering cross-section is clearly 
equal to the area S of the cross-section of the body by a plane perpendicular 
to the direction of the incident wave. The scattering through small angles 
(of the order of A//), however, constitutes diffraction from the edges of the 
body. We shall not pause here to expound the theory of this phenomenon, 
which is entirely analogous to that of the diffraction of light.f We shall only 
mention that, by Babinet's principle, the total intensity of diffracted sound 
is equal to the total intensity of reflected sound. Hence the diffraction 
part of the effective scattering cross-section is also equal to S, and the total 
cross-section is therefore 2S. 

PROBLEMS 

Problem 1 . Determine the effective cross-section for the scattering of a plane sound wave 
by a solid sphere of radius R small compared with the wavelength. 



t See The Classical Theory of Fields, §§7-7 to 7-9. 



§76 Scattering of sound 297 

Solution. The velocity at a given point in a plane wave is v = a cos wt. In the case of a 
sphere (see §11, Problem 1), the vector A is — £R 3 v. For the differential effective cross-section 
we obtain 

da = (1-f COS 0)2 do, 

9c 4 

where 8 is the angle between the direction of the incident wave and the direction of scattering. 
The scattered intensity is greatest in the direction 8 = -n, which is opposite to the direction 
of incidence. The total effective cross-section is 

a = (77r/9)( J R3 G> 2/ c 2)2. (1) 

Here (and also in Problems 3 and 4 below) it is assumed that the density p of the sphere 
is large compared with the density p of the gas ; if this were not so, it would be necessary 
to take account of the movement of the sphere by the pressure forces exerted on it by the 
oscillating gas. 

Problem 2. Determine the effective cross-section for the scattering of sound by a drop of 
fluid, taking into account the compressibility of the fluid and the motion of the drop caused 
by the incident wave. 

Solution. When the pressure of the gas surrounding the drop changes adiabatically by />', 
the volume of the drop is reduced by (V lp )(dp<>lc>p) s p', where p is the density of the drop. 
But (dp/dpo), is the square of the velocity of sound c in the fluid, and the pressure change in a 
plane wave is related to the velocity by p' = vcp, where p is the density of the gas. Thus 
the decrease in the volume of the drop is V vcpjc 2 p per unit time. In the expressions (76.2) 
and (76.3), we must now replace V vjc by the difference V {vlc—vcpjc 2 p^). Moreover, in 
the expression for A we must replace — v by the difference u— v, where u is the velocity 
acquired by the drop as a result of the action of the incident wave. For a sphere we have, using 
the results of §11, Problem 1, A = /? 3 v(P _ Po)/(2po + P)- Substituting these expressions, we 
have the effective cross-section 

co*R«l/ c*p \ po-p )* 

do- = {II — 3 cos 9 

9c4 \\ c 2p / 2po+p 



The total effective cross-section is 

4rrftAR6 // <fip \2 3(/> -/>) 2 



I \ Cn 2 po / 



+ 



9c* \\ coW (2p +p)2) 

Problem 3. Determine the effective cross-section for the scattering of sound by a solid 
sphere of radius R small compared with \/(vjw). The specific heat of the sphere is supposed 
so large that its temperature can be regarded as a constant. 

Solution. In this case we have to take into account the effect of the gas viscosity on the 
motion of the sphere, and the vector A must be modified as shown in §73, Problem 2. For 
RV(<oJv) < 1 we have A = —3iRvvJ2o). 

The thermal conductivity of the gas also results in scattering of the same order. Let 
TV - ' 6 " be the temperature variation at a given point in the sound wave. The temperature 
distribution near a sphere is (see §52, Problem 2) 

(for r = R we must have T' = 0). The amount of heat transferred from the gas to the sphere 
per unit time is (for RVHx) < 1) 9 = 4irR z k[&T I&r\ r=R = 4nR K T &-**>*. This transfer 
of heat results in a change in the volume of the gas, which can be taken to affect the scattering 
like a corresponding effective change in the volume of the sphere, V — — AirRxfiT' Q e~ im 
= — AirRx{y — l)v/c, where ]8 is the coefficient of thermal expansion of the gas and y = c p /c v ; 
we have used also formulae (63.13) and (77.2). 



298 Sound §77 

Taking account of both effects, we obtain the differential effective scattering cross-section 

da = (a)Rlc*)*\x(y - 1) - \v cos df do. 

The total effective cross-section is 

a = 477(ft> J R/c 2 ) 2 [x 2 (y-l)2 + |v2]. 

These formulae are valid only if the Stokes frictional force is small compared with the 
inertia force, i.e. tjR <^ Mco, where M = 4irR 3 p /3 is the mass of the sphere ; otherwise, the 
movement of the sphere by viscosity forces becomes important. 

Problem 4. Determine the mean force on a solid sphere which scatters a plane sound wave 
(A>#). 

Solution. The momentum transmitted per unit time from the incident wave to the sphere, 
i .e. the required force, is the difference between the momentum in the incident wave and the 
total momentum flux in the scattered wave. From the incident wave an energy flux acE 
is scattered, where E is the energy density in the incident wave ; the corresponding momen- 
tum flux is obtained by dividing by c, and is therefore oE . In the scattered wave, the momen- 
tum flux into the solid angle element do is E sc r 2 do = E da; projecting this on the direction 
of propagation of the incident wave (which is obviously the direction of the required force), 
and integrating over all angles, we obtain 

Eq COS 6 da. 

Thus the force on the sphere is 

F = Eo J(l-cos0)da. 

Substituting for da from Problem 1, we obtain F = UircoWE^fic*. 

§77. Absorption of sound 

The existence of viscosity and thermal conductivity results in the dissipa- 
tion of energy in sound waves, and the sound is consequently absorbed, 
i.e. its intensity progressively diminishes. To calculate the rate of energy 
dissipation i£mech> we use the following general arguments. The mechanical 
energy is just the maximum amount of work that can be done in passing 
from a given non-equilibrium state to one of thermodynamic equilibrium. 
As we know from thermodynamics,^ the maximum work is obtained when the 
transition is reversible (i.e. without change of entropy), and is then 
#mech = Bq — E(S), where Eo is the given initial value of the energy, and 
E(S) is the energy in the equilibrium state with the same entropy S as the 
system had initially. Differentiating with respect to time, we obtain 
$mech = —£(S) = -(dEIdS)S. The derivative of the energy with respect 
to the entropy is the temperature. Hence dE/dS is the temperature which 
the system would have if it were in thermodynamic equilibrium (with the 
given value of the entropy). Denoting this temperature by To, we therefore 
have .Cmech — — TqS. 



t See, for instance, Statistical Physics, §19. 



§77 Absorption of sound 299 

We use for 3 the expression (49.6), which gives the rate of change of the 
entropy due to both thermal conduction and viscosity. Since the temperature 
T varies only slightly through the fluid, and differs little from To, it can be 
taken outside the integral, and To can be written as T simply: 



k r f / OVi uVk OVi \ * 

&-- - Y /(«n«ID.dF- i ,J(- + --» ^) ir- 

-^J(divv)2dr. (77.1) 

This formula generalises formula (16.3) to the case of a compressible fluid 
which conducts heat. 

Let the #-axis be in the direction of propagation of the sound wave. Then 
v x = v cos(kx-wt), v y = v z = 0. The last two terms in (77.1) give 

-to + Q\(^\W = -Wto + frx? jsm*(kx-a>t)dV. 

We are, of course, interested only in the time average; taking this average, 
we have -& 2 (ji?+ £) . |^o 2 ^o, where Vq is the volume of the fluid. 

Next we calculate the first term in (77.1). The deviation V of the tem- 
perature in the sound wave from its equilibrium value is related to the 
velocity by formula (63.13), so that the temperature gradient is 

dTjdx = {^cTjc p )dvjdx = -(pcTlc p )voksm(kx-cot). 

For the time average of the first term in (77. 1) we obtain - KC 2 T(Pvo 2 k 2 Vol2c p 2 . 
Using the well-known thermodynamic formulae 

c p -c v = W(dp\d P ) T = T^{c v \c v \dp\d P ) s = TpWc v /c p , (77.2) 

we can rewrite this expression as — \k(\jc v — \jc p )k 2 vo 2 Vo. 

Collecting the above results, we find the mean value of the energy dissi- 
pation : 

imech= -lVo[(^ + + K(l/^-iy. (77.3) 

The total energy of the sound wave is 

E = \pvoW . (77.4) 

The damping coefficient derived in §25 for gravity waves gives the manner 
of decrease of the intensity with time. For sound, however, the problem is 
usually stated somewhat differently: a sound wave is propagated through a 
fluid, and its intensity decreases with the distance x traversed. It is evident 
that this decrease will occur according to a law e~ 2yx > and the amplitude will 
decrease as e~ yx , where the absorption coefficient y is defined by 

y = l^mechl/2^. (77.5) 



300 Sound §77 

Substituting here (77.3) and (77.4), we find the following expression for the 
sound absorption coefficient: 

r-^lto+o+ii-i)]- (77 - 6) 

We may point out that it is proportional to the square of the frequency of 
the sound.f 

This formula is applicable so long as the absorption coefficient determined 
by it is small: the amplitude must decrease relatively little over distances of 
the order of a wavelength (i.e. we must have ycjco <^ 1). The above deriva- 
tion is essentially founded on this assumption, since we have calculated 
the energy dissipation by using the expression for an undamped sound wave. 
For gases this condition is in practice always satisfied. Let us consider, for 
example, the first term in (77.6). The condition ycjco <^ 1 means that 
vcojc 2 <^ 1. It is known from the kinetic theory of gases, however, that the 
viscosity coefficient v for a gas is of the order of the product of the mean 
free path / and the mean thermal velocity of the molecules; the latter is of 
the same order as the velocity of sound in the gas, so that v ~ Ic. Hence we 
have 

vco/c 2 ~ lco/c ~ //A <^ 1, (77.7) 

since we know that / <^ A. The thermal-conduction term in (77.6) gives the 
same result, since x ~ v - 

In liquids, the condition of small absorption is always fulfilled when the 
problem of sound absorption, as stated here, is significant at all. The absorp- 
tion over one wavelength can become large only if the viscosity forces 
are comparable with the pressure forces which occur when the substance is 
compressed. In these conditions, however, the Navier-Stokes equation itself 
(with the viscosity coefficients independent of frequency) becomes invalid 
and a considerable dispersion of sound, due to processes of internal friction, 
occurs. % 

For absorption of sound, the relation between the wave number and the 
frequency can evidently be written 

k = co/c+iaco 2 , (77.8) 

where a denotes the coefficient of a> 2 in the absorption coefficient y = aco 2 . 



t M. A. Isakovich has shown that there must be a special absorption when sound is propagated 
in a two-phase system (an emulsion). Because of the different thermodynamic properties of the two 
components, their temperature changes during the passage of the sound wave will in general be 
different. The resulting heat exchange between the components leads to an additional absorption of 
sound. On account of the relative slowness of this heat exchange, a considerable dispersion of the 
sound takes place comparatively quickly. For detailed calculations see M. A. Isakovich, Zhurnal 
experimental' noi i teoreticheskoi fiziki 18, 907, 1948. 

J A special case where strong absorption is possible but can be discussed by the usual methods is 
that of a gas with a thermal conductivity which is unusually large compared with its viscosity, on 
account of effects such as radiative transfer at very high temperatures (see Problem 3). 



§77 Absorption of sound 301 

It is easy to see from this how the equation for a travelling sound wave must 
be modified in order to take absorption into account. To do so, we notice that, 
in the absence of absorption, the differential equation for (say) the pressure 
p' = p'{x-ct) can be written dp' fix = -{\jc)dp'jdt. The equation whose 
solution is e i(kx ~ wt \ with k given by (77.8), must clearly be 

v = _r_v + a ?v (77 . 9) 

dx c 8t dt 2 

If we replace t by t + x/c, this equation becomes 

dp' fix = adty'ldT*, 

i.e. a one-dimensional equation of thermal conduction. 

The general solution of this equation can be written (see §51) 

p'(x,r) = I f/o(r')exp[-(r'-T)2/te]dT', (77.10) 

where p'o(r) = p'(0, r). If the wave is emitted during a finite time interval, 
this expression becomes, at sufficiently large distances from the source, 

p'(x, r) = „ I exp( - r2/te) f p'o{r') dr'. (77.1 1) 

2y(7rax) J 

In other words, the wave profile at large distances is Gaussian. Its "width" 
is of the order of <\/(ax), i.e. it increases as the square root of the distance 
travelled by the wave, while the amplitude falls off inversely as y/x. Hence 
we at once conclude that the total energy of the wave decreases as l/y/x. 
It is easy to derive analogous formulae for spherical waves; to do so, we 
must use the fact that for such a wave 

jp'dt = 

(see §69). Instead of (77.11) we now have 

1 d exp(-T 2 /4«r) 
p'(r, t) = constant x 



r dr \/r 

or 

T 



p'(r,r) = constant x— exp(-r 2 /4ar). (77.12) 



fZ 



Strong absorption must occur when a sound wave is reflected from a solid 
wall (K. F. Herzfeld, 1938; B. P. Konstantinov, 1939). The reason for 
this is the following. In a sound wave not only the density and the pressure, 
but also the temperature, undergo periodic oscillations about their mean values. 
Near a solid wall, therefore, there is a periodically fluctuating temperature 
difference between the fluid and the wall, even if the mean fluid temperature is 



302 Sound §77 

equal to the wall temperature. At the wall itself, however, the temperatures 
of the wall and the adjoining fluid must be the same. As a result, a large 
temperature gradient is formed in a thin boundary layer of fluid, where the 
temperature changes rapidly from its value in the sound wave to the wall 
temperature. The presence of large temperature gradients, however, results 
in a large dissipation of energy by thermal conduction. For a similar reason, 
the fluid viscosity leads to strong absorption of sound when the wave is 
incident in an oblique direction. In this case the fluid velocity in the wave 
(in the direction of propagation) has a non-zero component tangential to the 
surface. At the surface itself, however, the fluid must completely "adhere". 
Hence a large tangential-velocity gradientf must occur in the boundary layer 
of fluid, resulting in a large viscous dissipation of energy (see Problem 1). 

PROBLEMS 

Problem 1. Determine the fraction of energy that is absorbed when a sound wave is 
reflected from a solid wall. The density of the wall is supposed so large that the sound does 
not penetrate it, and the specific heat so large that the temperature of the wall may be supposed 
constant. 

Solution. We take the plane of the wall as the plane x = 0, and the plane of incidence as 
the xy-plane. Let the angle of incidence (which equals the angle of reflection) be 0. The 
change in density in the incident wave at any given point on the surface (x = y = 0, say) 
is p\ = Ae*~ i(ot . The reflected wave has the same amplitude, so that p\ = p\ at the wall. 
The actual change in the fluid density, since both waves (incident and reflected) are propaga- 
ted simultaneously, is p' = 2Ae~ tu>t . The fluid velocity in the wave is given by v x = cp'iajp, 
v 2 = cp'jckjp. The total velocity on the wall, v = Vi+v 2 , is therefore v = v v = 2 A sin x 
ce-iut/p (or, more precisely, this is what the velocity is found to be when the correct boundary 
conditions at the wall in the presence of viscosity are not applied). The actual variation of the 
velocity v y along the wall is determined by formula (24.13), and the energy dissipation due to 
viscosity by formula (24.14), in which the above expression for v must be substituted for 
v e~ ia,t . 

The deviation T" of the temperature from its mean value (which is the temperature of the 
wall), if calculated without using the correct boundary conditions at the wall, would be found 
to be (see (63.13)) T — 2Ac i T^e- ib)t tc v p. In reality, however, the temperature distribution is 
determined by the equation of thermal conduction, with the boundary condition T" = for 
# = 0, and is accordingly given by a formula entirely similar to (24.13). 

On calculating the energy dissipation due to thermal conduction as the first term in formula 
(77.1), we obtain for the total energy dissipation per unit area of the wall 

•femech — 



[Vx(^-l) + Vvsin^]. 



P 

The mean energy flux density incident on unit area of the wall from the incident wave is 
cpVi 2 cos =*= (c a A 2 /2p) cos 0. Hence the fraction of energy absorbed on reflection is 

2V(2co) 



jvVsin204Vx(— -lYI. 



c cos 6 L \ c v 

This expression is valid only if its value is small (since in deriving it we have supposed the 
amplitudes of the incident and reflected waves to be the same). This condition means that 
the angle of incidence must not be too near Jir.J 



t The normal velocity component is zero at the boundary because of the boundary conditions, 
whether or not viscosity is present. 

J A calculation of the absorption of sound on reflection at any angle is given by B. P. Konstantinov, 
Zhurnal tekhnicheskol fiziki 9, 226, 1939. 



§77 Absorption of sound 303 

Problem 2. Determine the coefficient of absorption of sound propagated in a cylindrical 
pipe. 

Solution. The main contribution to the absorption is due to the presence of the walls. 
The absorption coefficient y is equal to the energy dissipated at the walls per unit time and 
per unit length of the pipe, divided by twice the total energy flux through a cross-section of the 
pipe. A calculation entirely similar to that given in Problem 1 leads to the result 



[ v , + vx(;-i)]. 



's/co 
7= y/2Rc 
where R is the radius of the pipe. 

Problem 3. Find the dispersion relation for sound propagated in a medium of very high 
thermal conductivity. 

Solution. In the presence of a large thermal conductivity the flow in a sound wave is not 
adiabatic. Hence, instead of the condition of constant entropy, we now have 

s=kAT'I p T, (1) 

which is the linearised form of equation (49.4) without the viscosity terms. As a second equa- 
tion we take 

P = AP', (2) 

which is obtained by eliminating v from equations (63.2) and (63.3). Taking as the funda- 
mental variables p' and T", we write p' and s' in the form 

p' = (dpldT) p T'+(8pldp) T p', s' = (dsldT) p T + (8sldp) T p\ 

We substitute these expressions in (1) and (2), and then seek T and p' in a form proportional 
to e i{ ~ kx ~ m) . The compatibility condition for the resulting two equations for p' and T can 
(by using various relations between the derivatives of thermodynamic quantities) be brought 
to the form 



(or ico\ 
ct 2 X i 



toy 



*4_*2 +_+—=<>, (3) 



XCs 2 



which gives the required relation between k and <a. We have here used the notation 
c s 2 = (8pldp)s, ct 2 = (dpl8p) T = c s 2 /y, 

where y = cjc v is the ratio of specific heats. 

In the limiting case of small frequencies (w <^ c 2 /x)» equation (3) gives 



CO C0 2 y / 1 1 \ 

k = — + i—^l , 

C s 2c s \Ct 2 C s 2 J 



which corresponds to the propagation of sound with the ordinary "adiabatic" velocity c t 
and a small absorption coefficient which is the second term in (77.6). This is as it should be, 
since the condition to <^ c 2 /x means that, during one period, heat can be transmitted only over 
a distance r>J V(xI Wl ) (cf. (51.7)) which is small compared with the wavelength c/co. 
In the opposite limiting case of large frequencies, we find from (3) 

co ct 
k = — + i- — -(c s 2 -c T 2 ). 
c T 2xc s 2 

In this case the sound is propagated with the "isothermal" velocity ct, which is always less 



304 Sound §78 

than c s . The absorption coefficient is again small compared with the reciprocal of the wave- 
length, and is independent of the frequency and inversely proportional to the thermal con- 
ductivity, f 

Problem 4. Determine the additional absorption, due to diffusion, of sound propagated 
in a mixture of two substances (I. G. Shaposhnikov and Z. A. Gol'dberg 1952). 

Solution. The mixture contains an additional source of absorption of sound because the 
temperature and pressure gradients occurring in the sound wave result in irreversible pro- 
cesses of thermal diffusion and barodiffusion (but there is evidently no mass-concentration 
gradient, and therefore no mass transfer). This absorption is given by the term 



(llT P D)(dn/8C) PtT j i*dV 



in the rate of change of entropy (58.13) ; we here denote the concentration by Cto distinguish 
it from c, the velocity of sound. The diffusion flux is 

i = - P D[(k T /T) grad T+ (k p /p) gradp], 

with k„ given by (58.10). A calculation similar to that given in §77, using various relations 
between the derivatives of thermodynamic quantities, leads to the result that there must be 
added to the expression (77.6) for the absorption coefficient a term 

Da 2 it dp \ k T l dp \ l dfi \ ) 2 

y D _ II I I II II 






2c p 2 (8ix/8C) Pi 

Problem 5. Determine the effective cross-section for the absorption of sound by a sphere 
of radius small compared with • v / ('V a, )• 

Solution. The total absorption is composed of the effects of the viscosity and thermal 
conductivity of the gas. The former is given by the work done by the Stokes frictional force 
when gas moving in a sound wave flows round a sphere ; as in §76, Problem 3, it is assumed that 
the sphere is not moved by this force. The effect of conductivity is given by the amount of 
heat q transferred from the gas to the sphere per unit time ( §76, Problem 3) : the energy dissi- 
pation when an amount of heat q is transferred, the temperature difference between the gas 
(far from the sphere) and the sphere being T, is qT'jT. The total effective absorption cross- 
section is found to be 



2ttR 
c 



H^- 1 )] 



§78. Second viscosity 

The second viscosity coefficient £ (which we shall call simply the second 
viscosity) is usually of the same order of magnitude as the viscosity coefficient 
7], There are, however, cases where £ can take values considerably exceeding 
rj. As we know, the second viscosity appears in processes which are accom- 
panied by a change in volume (i.e. in density) of the fluid. In compression 
or expansion, as in any rapid change of state, the fluid ceases to be in thermo- 
dynamic equilibrium, and internal processes are set up in it which tend to 



f The second root of equation (3), which is quadratic in k 2 , corresponds to "thermal waves" which 
are rapidly damped with increasing x. In the limit a>x <^ c 2 this root gives 

* = V(Wx) = (i+0V(W2x), 

in agreement with (52.17). In the case cax ^> c 2 we have 

k = (1 +i)\/(cocv/2xc p ). 



§78 Second viscosity 305 

restore this equilibrium. These processes are usually so rapid (i.e. their relaxa- 
tion time is so short) that the restoration of equilibrium follows the change 
in volume almost immediately unless, of course, the rate of change of volume 
is very large. 

It may happen, nevertheless, that the relaxation times of the processes of 
restoration of equilibrium are long, i.e. they take place comparatively slowly. 
For instance, if we are concerned with a liquid or gas which is a mixture of 
substances between which a chemical reaction occurs, there is a state of chemi- 
cal equilibrium, characterised by the concentrations of the substances in 
the mixture, for any given density and temperature. If, for example, we 
compress the fluid, the state of equilibrium is destroyed, and a reaction 
begins, as a result of which the concentrations of the substances tend to take 
the equilibrium values corresponding to the new density and temperature. 
If this reaction is not rapid, the restoration of equilibrium occurs relatively 
slowly and does not immediately follow the compression. The latter process 
is then accompanied by internal processes which tend towards the equilibrium 
state. But the processes which establish equilibrium are irreversible; they 
increase the entropy, and therefore involve energy dissipation. Hence, if the 
relaxation time of these processes is long, a considerable dissipation of energy 
occurs when the fluid is compressed or expanded, and, since this dissipation 
must be determined by the second viscosity, we reach the conclusion that 
£ is large.f 

The intensity of the dissipative processes, and therefore the value of £, 
depend of course on the relation between the rate of compression or expansion 
and the relaxation time. If, for example, we have compression or expansion 
due to a sound wave, the second viscosity will depend on the frequency of the 
wave. Thus the second viscosity is not just a constant characteristic of the 
material concerned, but depends on the frequency of the motion in which it 
appears. The dependence of £ on the frequency is called its dispersion. 

The following general method of discussing all these phenomena is due to 
L. I. Mandel'shtam and M. A. Leontovich (1937). Let £ be some physical 
quantity characterising the state of a body, and |o its value in the equilibrium 
state; |o is a function of density and temperature. For instance, in fluid mix- 
tures | may be the concentration of one component, and then |o is the con- 
centration in chemical equilibrium. 

If the body is not in equilibrium, £ will vary with time, tending to the value 
|o- In states close to equilibrium the difference £ — £o is small, and we can 
expand the rate of change £ of £ in a series of powers of this difference. 
The zero-order term is absent, since £ must be zero in the equilibrium state, 
i.e. when £ = £o- Hence, as far as the first-order term, we have 

£= -(£-&)/t. (78.1) 

The proportionality coefficient must be negative, since otherwise £ would not 



f A slow process which results in a large £ is often the transfer of energy from translator/ degrees 
of freedom of a molecule to vibrational (intramolecular) degrees of freedom. 



306 Sound §78 

tend to a finite limit. The positive constant t is of the dimensions of time, 
and may be regarded as the relaxation time for the process in question; the 
greater is t, the more slowly the approach to equilibrium takes place. 

In what follows we shall consider processes in which the fluid is subjected 
to a periodic adiabaticf compression and expansion, so that the variable part 
of the density (and of the other thermodynamic quantities) depends on the 
time through a factor e- i<ot ; we are considering a sound wave in the fluid. 
Together with the density and other quantities, the position of equilibrium 
also varies, so that £ can be written as £ = £oo+fo', where £00 is the 
constant value of £ corresponding to the mean density, and £o' is a periodic 
part, proportional to e~ iut . Writing the true value £ in the form g = £ o+ I', 
we conclude from equation (78.1) that £' also is a periodic function of time, 
related to £ o' by 

£' = So'Kl-ian). (78.2) 

Let us calculate the derivative of the pressure with respect to the density 
for the ptocess in question. The pressure must now be regarded as a function 
of the density and of the value of £ in the state concerned, and also of the 
entropy, which we suppose constant and, for brevity, omit. Then 

dpi dp = (dpldp\+(dpldt) p 8£/d P . 

In accordance with (78.2), we substitute here 

% %' 1 a&' 1 a& 



obtaining 



dp dp I — tear dp 1 — icoT dp 



»._L|(*) + (»)*.U»)]. 

dp l-ton\\dp/ i \d£/ p dp \8pf s l 

The sum (dpl8p)g + (dpld$) p d(joldp is just the derivative of p with respect to 
p for a process which is so slow that the fluid remains in equilibrium; denoting 
it by (dp}dp) m , we have finally 



(78.3) 



dp 1 — iorr |_ \ dp} eq \ dp J g_ 

Next, let po be the pressure in a state of thermodynamic equilibrium; 
Po is related to the other thermodynamic quantities by the equation of state 
of the fluid, and is entirely determined when the density and entropy are 
given. The pressure p in a non-equilibrium state, however, differs from po, 
and is a function of £ also. If the density is adiabatically increased by 8p, 
the equilibrium pressure changes by Spo = {dpjdp) eq hp i while the total 
increase in the pressure is (dp/8p)Sp, with dp/dp given by formula (78.3). 



t The change in the entropy (in states close to equilibrium) is of the second order of smallness. 
Hence, to this order of accuracy, we can speak of an adiabatic process. 



§78 Second viscosity 307 

Hence the difference p — />o between the true pressure and the equilibrium 
pressure, in a state where the density is p + 8p, is 

I dp \dp /eqj 1 — icor \_\ dp / eq \ dp / ^J 

We are here interested in the density changes due to the motion of the 
fluid. Then 8p is related to the velocity by the equation of continuity, 
which we write in the form d(8p)jdt+p div v = 0, where d/dt denotes the 
total time derivative. In a periodic motion we have d(8p)/dt = — ico8p, 
and therefore 8p = (pjico) div v. Substituting this expression in (78.3a), 
we obtain 

P-Po = t^t- (co 2 -cJ) divv, (78.4) 

1— 10)T 

where we have used the notation 

co 2 = (dpldp)^ cj = (dpld P ) g , (78.5) 

the significance of which will be explained below. 

In order to relate these expressions to the viscosity of the fluid, we write 
down the stress tensor aye. In this tensor the pressure appears in the term 
— p8iic. Subtracting the pressure po determined by the equation of state, we 
find that in a non-equilibrium state o^ contains an additional term 

rp 
-{p-po)8ik = — L . — (cj-co^ucdivv. 

\—l(DT 

Comparing this with the general expression (15.2) and (15.3) for the stress 
tensor, in which div v appears in the term £ div v, we conclude that the 
presence of slow processes tending to establish equilibrium is macroscopically 
equivalent to the presence of a second viscosity given by 

t = Tp(cJ-c *)l(l-icoT). (78.6) 

These processes do not affect the ordinary viscosity 7). For processes so slow 
that cot <^ 1, £ is 

£o = t P (cJ-c 2); (78.7) 

it increases with the relaxation time r, in accordance with what was said 
above. For large frequencies, £ depends on the frequency, i.e. it exhibits 
dispersion. 

Let us now consider the question of how the presence of processes with 
large relaxation times (for definiteness, we shall speak of chemical reactions) 
affects the propagation of sound in a fluid. To do so, we might start from 
the equation of motion of a viscous fluid, with £ given by formula (78.6). 
It is simpler, however, to consider a motion in which viscosity is neglected 
but the pressure p is given by the above formulae instead of by the equation 
of state. The general relations which we obtained in §63 then remain formally 
applicable. In particular, the wave number and the frequency are still 



308 Sound §78 

related by k = co/c, where c = ^/(dpjdp), and the derivative dp/dp is now 
given by (78.3) ; the quantity c, however, no longer denotes the velocity of 
sound, being complex. Thus we obtain 

k = coV[(1-^)/(c 2 -Coo 2 *wt)]. (78.8) 

The "wave number" given by this formula is complex. The meaning of 
this fact is easily seen. In a plane wave, all quantities depend on the co- 
ordinate x (the #-axis being in the direction of propagation) through a factor 
e ikx . Writing k in the form k = ki + ik 2 with k\, k 2 real, we have e ikx = 
e i\x e -Jc 2 x t i #e besides the periodic factor e ik i x we have a damping factor e~ k 2 x 
(&2 mustj of course, be positive). Thus the complex nature of the wave 
number formally expresses the fact that the wave is damped, i.e. there is 
absorption of sound. The real part of the complex wave number gives the 
variation in phase of the wave with distance, and the imaginary part is the 
absorption coefficient. 

It is not difficult to separate the real and imaginary parts of (78.8). In 
the general case of arbitrary co the expressions for k\ and k 2 are rather cum- 
bersome, and we shall not write them out here. It is important that k\ 
is a function of the frequency (as is £2)- Thus, if chemical reactions can occur 
in the fluid, the propagation of sound at sufficiently high frequencies is 
accompanied by dispersion. 

In the limiting case of low frequencies (cot <^ 1), formula (78.8) gives 
to a first approximation k = cojco, corresponding to the propagation of sound 
with velocity cq. This is as it should be, of course: the condition cot <4 1 
means that the period 1/co of the sound wave is large compared with the 
relaxation time, i.e. the establishment of chemical equilibrium follows the 
variations of density in the sound wave, and the velocity of sound is deter- 
mined by the equilibrium value of the derivative dp/ dp. In the second approxi- 
mation We have 

co ico 2 T 
k = - + -—(cJ-co*), (78.9) 

CO lC{f 

i.e. damping occurs, with a coefficient proportional to the square of the fre- 
quency. Using (78.7), we can write the imaginary part of k in the form 
k% = co 2 £o/2pco 3 ; this agrees with the ^-dependent part of the absorption 
coefficient y as given by (77.6), which was obtained without taking account 
of the dispersion. 

In the opposite limiting case of high frequencies (cot > 1), we have 
in the first approximation k = cojc^, i.e. the propagation of sound with 
velocity c ro — again a natural result, since for cot p 1 we can suppose that 
no reaction occurs during a single period, and the velocity of sound must 
therefore be determined by the derivative (dpjdp)g taken at constant concen- 
tration. The second approximation gives 

k = - + i^—±. (78.10) 

c^ 2tc* 



§78 Second viscosity 309 

The damping coefficient is independent of the frequency. As we go from 
co <^ 1/r to o) ^> 1/t, this coefficient increases monotonically to the constant 
value given by formula (78.10). It should be noted that the quantity A 2 /&i, 
which represents the amount of absorption over a distance of one wavelength, 
is small in both limiting cases {fa\k\ < 1) ; it has a maximum at some inter- 
mediate frequency, namely co = -v/OVO/ 7 "- 
It is seen from (78.7) (e.g.) that 

Coo > co, (78.11) 

since we must have £ > 0. The same result can be obtained by simple 
arguments based on Le Chatelier's principle. Let us suppose that the 
volume of the system is reduced, and the density increased, by some external 
agency. The system is thereby brought out of equilibrium, and according 
to Le Chatelier's principle processes must begin which tend to reduce the 
pressure. This means that dp/ dp will decrease, and, when the system returns 
to equilibrium, the value of dpi dp = c 2 will be less than in the non-equili- 
brium state. 

In deriving all the above formulae we have assumed that there is only a 
single slow internal process of relaxation. Cases are also possible where 
several different such processes occur simultaneously. All the formulae can 
easily be generalised to cover such cases. Instead of a single quantity £, 
we now have several quantities £i, £2, ... which characterise the state of the 
system, and a corresponding series of relaxation times ti, T2, .... We choose 
the quantities £ w in such a way that each of the derivatives | n depends 
only on the corresponding |», i.e. so that 

in = -tf»-f«o)/T„. (78.12) 

Calculations entirely similar to the above then give 

& = cJ+J^ a n l(l-icoT n ), (78.13) 

n 

where c ro 2 = {dpjdp)^ and the constants a n are 

a n = (Bpld€nW£nldp)e+ ( 78 - 14 ) 

If there is only one quantity £, formula (78.13) becomes (78.3), as it should. 



CHAPTER IX 

SHOCK WAVES 

§79. Propagation of disturbances in a moving gas 

When the velocity of a fluid in motion becomes comparable with or exceeds 
that of sound, effects due to the compressibility of the fluid become of prime 
importance. Such motions are in practice met with in gases. The dynamics 
of high-speed flow is therefore usually called gas dynamics. 

It should be mentioned first of all that, in gas dynamics, the Reynolds 
numbers involved are almost always very large. For the kinematic viscosity 
of a gas is, as we know from the kinetic theory of gases, of the order of the 
mean free path / of the molecules multiplied by the mean velocity of their 
thermal motion; the latter is of the same order as the velocity of sound, so that 
v ~ cl. If the characteristic velocity in a problem of gas dynamics is also of 
the order of c, then the Reynolds number R ~ Lc\v ~ Ljl, i.e. it is deter- 
mined by the ratio of the dimension L to the mean free path /, which we know 
is very large.f As always occurs when R is very large, the viscosity has an 
important effect on the motion of the gas only in a very small region, and in 
what follows we shall (except where the contrary is specifically stated) regard 
the gas as an ideal fluid. 

The flow of a gas is entirely different in nature according as it is subsonic 
or supersonic, i.e. the velocity is less than or greater than that of sound. 
One of the most important distinctive features of supersonic flow is the fact 
that there can occur in it what are called shock waves, whose properties we 
shall examine in detail in the following sections. Here we shall consider 
another characteristic property of supersonic flow, relating to the manner of 
propagation of small disturbances in the gas. 

If a gas in steady motion receives a slight perturbation at any point, the 
effect of the perturbation is subsequently propagated through the gas with 
the velocity of sound (relative to the gas itself). The rate of propagation of 
the disturbance relative to a fixed system of co-ordinates is composed of 
two parts: firstly, the perturbation is "carried along" by the gas flow with 
velocity v and, secondly, it is propagated relative to the gas with velocity c 
in any direction n. Let us consider, for simplicity, a uniform flow of 
gas with constant velocity v, subjected to a small perturbation at some 
point O (fixed in space). The velocity v+cn with which the perturbation 
is propagated from O (relative to the fixed system of co-ordinates) has 
different values for different directions of the unit vector n. We obtain 



f We shall not consider the problem of the motion of bodies in very rarefied gases, where the 
mean free path of the molecules is comparable with the dimension of the body. This problem is in 
essence not one of fluid dynamics, and must be examined by means of the kinetic theory of gases. 

310 



§79 Propagation of disturbances in a moving gas 311 

all its possible values by placing one end of the vector v at the point O 
and drawing a sphere of radius c centred at the other end. The vectors from 
O to points on this sphere give the possible magnitudes and directions of the 
velocity of propagation of the perturbation. Let us first suppose that v < c . 
Then the vector v + cn can have any direction in space (Fig. 40a). That 
is, a disturbance which starts from any point in a subsonic flow will eventually 
reach every point in the gas. If, on the other hand, v > c, the direction of the 
vector v + cn can lie, as we see from Fig. 40b, only in a cone with its vertex at 
O, which touches the sphere with its centre at the other end of the vector v. 
If the aperture of the cone is 2a, then, as is seen from the figure, 

sin a = cjv. (79.1) 





Fig. 40 



Thus a disturbance starting from any point in a supersonic flow is propagated 
only downstream within a cone whose aperture is the smaller, the smaller the 
ratio cjv. A disturbance starting from O does not affect the flow outside 
this cone. 

The angle a determined by equation (79.1) is called the Mach angle. 
The ratio v/c itself, which often occurs in gas dynamics, is the Mach number M : 

M = v/c. (79.2) 

The surface bounding the region reached by a disturbance starting from a 
given point is called the Mach surface or characteristic surface. 

In the general case of an arbitrary steady flow, the Mach surface is not a 
cone throughout the volume. However, it can be asserted that, as before, this 
surface cuts the streamline through any point on it at the Mach angle. The 
value of the Mach angle varies from point to point with the velocities v and c. 
It should be emphasised here, incidentally, that, in flow with high velocities, 
the velocity of sound is different at different points : it varies with the ther- 
modynamic quantities (pressure, density, etc.) of which it is a function. f 
The velocity of sound as a function of the co-ordinates is sometimes called 
the local velocity of sound. 



t In the discussion of sound waves given in Chapter VIII, the velocity of sound could be regarded 
as constant. 



312 Shock Waves §79 

The properties of supersonic flow described above give it a character quite 
different from that of subsonic flow. If a subsonic gas flow meets any 
obstacle (if, for instance, it flows past a body), the presence of this obstacle 
affects the flow in all space, both upstream and downstream; the effect 
of the obstacle is zero only asymptotically at an infinite distance from it. 
A supersonic flow, however, is incident "blindly" on an obstacle; the effect 
of the latter extends only downstream,f and in all the remaining part of 
space upstream the gas flows as if the obstacle were absent. 

In the case of steady plane flow of a gas, the characteristic surfaces can be 
replaced by characteristic lines (or simply characteristics) in the plane of the 
flow. Through any point O in this plane there pass two characteristics (AA' 
and BB' in Fig. 41), which intersect the streamline through this point at the 
Mach angle. The downstream branches OA and OB of the characteristics 
maybe said to leave the point O; they bound the region AOB of the flow 
where perturbations starting from O can take effect. The branches B'O 
and A'O may be said to reach the point O; the region A' OB' between them 
is that which can affect the flow at O. 




The concept of characteristics (surfaces in the three-dimensional case) 
has also a somewhat different aspect. They are rays along which disturbances 
are "propagated" which satisfy the conditions of geometrical acoustics. If, 
for example, a steady supersonic gas flow meets a fairly small obstacle, then a 
steady perturbation of the gas flow will be found along the characteristics 
which leave this obstacle. The same result was reached in §67 from a study 
of the geometrical acoustics of moving media. 

When we speak of a perturbation of the state of the gas, we mean a slight 
change in any of the quantities characterising its state : the velocity, pressure, 



t To avoid misunderstanding, we should mention that, if a shock wave is formed in front of the 
obstacle, this region is somewhat enlarged (see §114). 



§80 Steady flow of a gas 313 

density, etc. The following remark should be made on this point. Pertur- 
bations in the values of the entropy of the gas (for constant pressure) and of 
its vorticity are not propagated with the velocity of sound. These perturba- 
tions, once having arisen, do not move relative to the gas; relative to a fixed 
system of co-ordinates they move with the gas at the velocity appropriate to 
each point. For the entropy, this is an immediate consequence of the law of 
conservation (in an ideal fluid), 

ds/dt = ds/dt+v-grads = 0, 

which shows that the entropy of any given volume element in the gas remains 
constant as the element moves about, i.e. each value of s moves with the 
point to which it belongs. The same result for the vorticity follows from the 
conservation of circulation. 

Thus we can say that, for perturbations of entropy and vorticity, the 
characteristics are the streamlines. This, of course, does not affect the general 
validity of the statements made above about regions of influence, since 
they were based only on the existence of a maximum velocity of propagation 
(that of sound) of disturbances relative to the gas itself. 

§80. Steady flow of a gas 

We can obtain immediately from Bernoulli's equation a number of general 
results concerning adiabatic steady flow of a gas. The equation is, for steady 
flow, w + \v 2 = constant along each streamline; if we have potential flow, 
then the constant is the same for every streamline, i.e. at every point in the 
fluid. If there is a point on some streamline at which the gas velocity is zero, 
then we can write Bernoulli's equation as 

w+±v 2 = wo, (80.1) 

where wq is the value of the heat function at the point where v = 0. 

The equation of conservation of entropy for steady flow is v«grads 
= vdsjdl = 0, i.e. 5 is constant along each streamline. We can write this in a 
form analogous to (80.1): 

s = s . (80.2) 

We see from equation (80.1) that the velocity v is greater at points where 
the heat function w is smaller. The maximum value of the velocity (on 
the streamline considered) is found at the point where w is least. For con- 
stant entropy, however, we have dw = dpjp\ since p > 0, the differentials 
dw and dp have like signs, and therefore w and p vary in the same sense. 
We can therefore say that the velocity increases along a streamline when the 
pressure decreases, and vice versa. 

The smallest possible values of the pressure and the heat function (in 
adiabatic flow) are obtained when the absolute temperature T = 0. The 
corresponding pressure is p = 0, and the value of w for T = can be 
arbitrarily taken as the zero of energy; then w = for T = 0. We can 



314 



Shock Waves 



§80 



now deduce from (80.1) that the greatest possible value of the velocity (for 
given values of the thermodynamic quantities at the point where v = 0) is 

%ax = V(2wo). (80.3) 

This velocity can be attained when a gas flows steadily out into a vacuum.f 
Let us now consider how the mass flux density j = pv varies along a 
streamline. From Euler's equation (vgrad)v = -(l//>)grad/>, we find 
that the relation v dv = dpjp between the differentials dv and dp holds 
along a streamline. Putting dp = c 2 dp, we have 



dp/dv = —pvjc 2 
and, substituting in d(pv) = p dv + v dp, we obtain 

d(pv){dv = p(l—v 2 /c 2 ). 



(80.4) 



(80.5) 



tf. 



0-79 
O-50 
0-25 



















































































0-25 0-50 0-75 



TOO 



1-25 
Y/ 
Fig. 42 



1-50 



1-75 2-00 2-25 2-50 



From this we see that, as the velocity increases along a streamline, the 
mass flux density increases as long as the flow remains subsonic. In the super- 
sonic range, however, the mass flux density diminishes with increasing 
velocity, and vanishes together with p when v — v ma , x (Fig- 42). This im- 
portant difference between subsonic and supersonic steady flows can be simply 
interpreted as follows. In a subsonic flow, the streamlines approach in the 
direction of increasing velocity. In a supersonic flow, however, they diverge 
in that direction. 

The flux j has its maximum value j+ at the point where the gas velocity is 
equal to the local velocity of sound : 

j* = p*c*, (80.6) 

where the asterisk suffix indicates values corresponding to this point. The 



t In reality, of course, when there is a sharp fall in temperature the gas must condense and form a 
two-phase "fog". This, however, does not essentially affect the results given. 



§80 Steady flow of a gas 315 

velocity v* = c # is called the critical velocity. In the general case of an arbi- 
trary gas, the critical values of quantities can be expressed in terms of their 
values at the point v = 0, by solving the simultaneous equations 

s* = % «> # + |c* 2 = w . (80.7) 

It is evident that, whenever M = vjc < 1, we have also vfc* < 1, and 
if M > 1 then vjc^ > 1. Hence the ratio M* = v\c % serves in this case as a 
criterion analogous to M, and is more convenient, since c % is a constant, 
unlike c, which varies along the stream. 

In applications of the general equations of gas dynamics, the case of a 
perfect gas is of particular importance. For a perfect gas we know from 
thermodynamics all the relations between the various thermodynamic 
quantities, and these relations are very simple. This makes it possible to give 
a complete solution of the equations of gas dynamics in many cases. 

We shall give here, for reference, the relations between the various thermo- 
dynamic quantities for a perfect gas, since they will often be needed in what 
follows. We shall always assume (unless otherwise stated) that the specific 
heat of a perfect gas is independent of temperature. 

The equation of state for a perfect gas is 

pV = pip = RT/p, (80.8) 

where R = 8-314 xlO 7 erg/deg is the gas constant, and /x the molecular 
weight of the gas. The velocity of sound in a perfect gas is, as shown in §63, 
given by 

c* = yRT/fi = yp/p, (80.9) 

where we have introduced the constant ratio of specific heats y = c v Jc v , 
which always exceeds unity; for monatomic gases y = 5/3, and for diatomic 
gases y — 7/5, at ordinary temperatures. 

The internal energy of a perfect gas is, apart from an unimportant additive 
constant, 

c = c v T = pV\iy- 1) = c*ly{y- 1). (80.10) 

For the heat function we have the analogous formulae 

to = CpT = ypV/(y- 1) = c^{y- 1). (80.11) 

Here we have used the well-known relation c p — c v = R/p,. Finally, the 
entropy of the gas is 

s = c v log(pl P r) = c v log(pVr/p). (80.12) 

Let us now investigate steady flow, applying the general relations pre- 
viously obtained to the case of a perfect gas. Substituting (80.11) in (80.3), 
we find that the maximum velocity of steady flow is 

%ax = coV[2/(y-l)l (80.13) 



316 Shock Waves §80 

For the critical velocity we obtain from the second equation (80.7) 



+ \c£ = Wq = 



y-\ y-\ 

whencej* 

c* = W[2/(y+l)]. (80.14) 

Bernoulli's equation (80.1), after substitution of the expression (80.11) 
for the heat function, gives the relation between the temperature and the 
velocity at any point on the streamline ; similar relations for the pressure and 
density can then be obtained directly by means of Poisson's adiabatic equa- 
tion: 

(80.15) 



Thus 


p = P o(777o)i/Cr-i), p = 
we obtain the important results 

r-r.[i-« y -i)J-r.(i-r 

r v 2 -i l/(r-D / 

p = po[i-Kr-i)^-J =po(i 
/>=/»o[i-Kr-i)-^-J =Po(i 


Po(p/po) y . 

-1 V 2 \ 


• 




+ i c.*r 

y—\ v 2 

y+i a 

y— 1 v 2 
y+\ c 2 


\l/(y-l) 

■) • 

.)■•-] 



(80.16) 



It is sometimes convenient to use these relations in a form which gives the 
velocity in terms of other quantities : 

^ = iLALf, _(£pn . ^th-iJLfX (8 o.i7) 

y-1 po L \po/ J y—lpol \ po 1 J 

We may also give the relation between the velocity of sound and the 
velocity v : 

C 2 = co 2 -±(y- \)v 2 = ±(y + l)*. a -i(y- l)^ 2 . (80.18) 

Hence we find that the numbers M and M* are related by 

y+1 
M*2 = L . ; (80.19) 

y-l+2/M2' V ; 

when M varies from to oo, M* 2 varies from to (y+ l)/(y— 1). 

Finally^ we may give expressions for the critical temperature, pressure and 
density: they are obtained by putting v = c% in formulae (80.16)$: 



f Fig. 42 shows the ratio jlj# as a function of vjc^ for air (y = 1-4, f ma x = 2-45^). 
J For air, e.g., (y = 1-4) 

c* = 0-913co, p* = 0-528/>o, />* = 0-634/j , T m = 0-833 T . 



§81 Surfaces of discontinuity 317 



T* = 27o/(y+l), 

/ 2 \r/(r-i) 

/>* = po — — - 
\y+l 



(80.20) 



In conclusion, it should be emphasised that the results derived above are 
valid only for flow in which shock waves do not occur. When shock waves are 
present, equation (80.2) does not hold ; the entropy of the gas increases when 
a streamline passes through a shock wave. We shall see, however, that 
Bernoulli's equation (80.1) remains valid even when there are shock waves, 
since w + %v 2 is a quantity which is conserved across a surface of discon- 
tinuity (§82); formula (80.14), for example, therefore remains valid also. 

PROBLEM 

Express the temperature, pressure and density along a streamline in terms of the Mach 
number. 

Solution. Using the formulae obtained above, we find 

T /T = 1 +Ky- 1)M 2 , po/p = [1 +1(7- l)M2]7/(7-i), 

poIp = [l+Mr-^M 2 ] 1 ^- 1 ). 

§81. Surfaces of discontinuity 

In the preceding chapters we have considered only flows such that all 
quantities (velocity, pressure, density, etc.) vary continuously. Flows are 
also possible, however, for which discontinuities in the distribution of these 
quantities occur. 

A discontinuity in a gas flow occurs over one or more surfaces ; the quan- 
tities concerned change discontinuously as we cross such a surface, which is 
called a surface of discontinuity. In non-steady gas flow the surfaces of dis- 
continuity do not in general remain fixed; here it should be emphasised, 
however, that the rate of motion of these surfaces bears no relation to the 
velocity of the gas flow itself. The gas particles in their motion may cross a 
surface of discontinuity. 

Certain boundary conditions must be satisfied on surfaces of discontinuity. 
To formulate these conditions, we consider an element of the surface and use 
a co-ordinate system fixed to this element, with the #-axis along the normal.f 

Firstly, the mass flux must be continuous : the mass of gas coming from 
one side must equal the mass leaving the other side. The mass flux through 
the surface element considered is pv x per unit area. Hence we must have 
pivix = P2V2x, where the suffixes 1 and 2 refer to the two sides of the surface 
of discontinuity. 



f If the flow is not steady, we consider an element of the surface during a short interval of time. 



318 Shock Waves §81 

The difference between the values of any quantity on the two sides of 
the surface will be denoted by enclosing it in square brackets; for example, 
[pv x ] == piviz— p2V2z, and the condition just derived can be written 

[f>vx] = 0. (81.1) 

Next, the energy flux must be continuous. The energy flux is given by 
(6.3). We therefore obtain the condition 

[pvafttP + w)] = 0. (81.2) 

Finally, the momentum flux must be continuous, i.e. the forces exerted 
on each other by the gases on the two sides of the surface of discontinuity 
must be equal. The momentum flux per unit area is (see §7) pm + pViVtfik. 
The normal vector n is along the rc-axis. The continuity of the ^-component 
of the momentum flux therefore gives the condition 

Ip+pvJ] = 0, (81.3) 

while that of the y and z components gives 

IpVaPy] = 0, \pv x v z ] = 0. (81.4) 

Equations (81.1)— (81.4) form a complete system of boundary conditions 
at a surface of discontinuity. From them we can immediately deduce the 
possibility of two types of surface of discontinuity. 

In the first type, there is no mass flux through the surface. This means 
that pivix = p$V2x = 0. Since pi and p2 are not zero, it follows that v\ x 
= V2x = 0. The conditions (81.2) and (81.4) are then satisfied, and the con- 
dition (81.3) gives pi = P2. Thus the normal velocity component and the 
gas pressure are continuous at the surface of discontinuity: 

vix = v 2x = 0, \p] = 0, (81.5) 

while the tangential velocities v y , v z and the density (as well as the other 
thermodynamic quantities except the pressure) may be discontinuous by any 
amount. We call this a tangential discontinuity. 

In the second type, the mass flux is not zero, and v\ x and V2x are therefore 
also not zero. We then have from (81.1) and (81.4) 

[vy] = 0, [v z ] = 0, (81.6) 

i.e. the tangential velocity is continuous at the surface of discontinuity. 
The pressure, the density (and the other thermodynamic quantities) and the 
normal velocity, however, are discontinuous, their discontinuities being 
related by (81.1) — (81.3). In the condition (81.2) we can cancel pv x by 
(81.1), and replace v 2 by v x 2 since v y and v z are continuous. Thus the 
following conditions must hold at the surface of discontinuity in this case : 

b*>x] = 0, \ 

[W + «7] = 0, (81.7) 

[p + pv x *] = 0. J 

A discontinuity of this kind is called a shock wave, or simply a shock. 



§82 The shock adiabatic 319 

If we now return to the fixed co-ordinate system, we must everywhere 
replace v x by the difference between the gas velocity component v n normal 
to the surface of discontinuity and the velocity u of the surface itself, which is 
defined to be normal to the surface : 

v x = v n — u. (81.8) 

The velocities v and « are taken in the fixed system. The velocity v% is 
the velocity of the gas relative to the surface of discontinuity; we can also 
say that — v x = u — v n is the rate of propagation of the surface relative to 
the gas. It should be noticed that, if v x is discontinuous, this velocity has 
different values relative to the gas on the two sides of the surface. 

We have already discussed (in §30) tangential discontinuities, at which the 
tangential velocity component is discontinuous, and we showed that, in an 
incompressible fluid, such discontinuities are absolutely unstable and must 
result in a turbulent region. A similar investigation for a compressible fluid 
shows that the same instability occurs, for any velocities. 

A particular "degenerate" case of tangential discontinuity is that where the 
velocity is continuous, but not the density (and therefore the other thermo- 
dynamic quantities, except the pressure). The above remarks on instability 
do not relate to discontinuities of this kind. 

§82. The shock adiabatic 

Let us now investigate shock waves in detail. We have seen that, in this 
type of discontinuity, the tangential component of the gas velocity is con- 
tinuous. We can therefore take a co-ordinate system in which the surface 
element considered is at rest, and the tangential component of the gas velocity 
is zero on both sides.f Then we can write the normal component v x as v 
simply, and the conditions (81.7) take the form 

pivi = p 2 v 2 =j, (82.1) 

pi + pivi 2 = pz + P2^2 2 , (82.2) 

Wl + |©l 2 = W2 + |^2 2 , (82.3) 

where j denotes the mass flux density at the surface of discontinuity. In 
what follows we shall always take j positive, with the gas going from side 1 
to side 2. That is, we call gas 1 the one into which the shock wave moves, 
and gas 2 that which remains behind the shock. We call the side of the shock 
wave towards gas 1 the front of the shock, and that towards gas 2 the back. 
We shall derive a series of relations which follow from the above condi- 
tions. Using the specific volumes V\ = l//>i, V2 = I//02, we obtain from 
(82.1) 

vi = jV lt v 2 = jV 2 (82.4) 



t This co-ordinate system is used everywhere in §§82-85, 87, 88. 

A shock wave at rest is called a compression discontinuity. If the shock is perpendicular to the 
direction of flow, we have a normal shock, otherwise an oblique shock. 



320 Shock Waves 

and, substituting in (82.2), 



pi+pVi = p2+j 2 V 2 , 



§82 



(82.5) 



or 



j 2 = (p2-pi)l(V 1 -V 2 ). (82.6) 

This formula, together with (82.4), relates the rate of propagation of a shock 
wave to the pressures and densities of the gas on the two sides of the surface. 




Fig. 43 



Since p is positive, we see that either p% > pi, V\ > V2, or p% < pi, 
V\ < V2', we shall see below that only the former case can actually occur. 

We may note the following useful formula for the velocity difference 
©1 — ^2. Substituting (82.6) in V1 — V2 = j{V\— V2), we obtainf 

Vl -V2 = V[(p2-piWi-V2)]. (82.7) 

Next, we Write (82.3) in the form 

^l+i/W = W2 + hpV 2 2 (82.8) 

and, substituting/ 2 from (82.6), obtain 

W!-w 2 + !( V x + V 2 ){p 2 -pi) = 0. (82.9) 

If we replace the heat function why e+pV, where e is the internal energy, we 
can write this relation as 



ei-e2 + l(V 1 -V 2 ){pi+p2) = 0. 



(82.10) 



These relations hold between the thermodynamic quantities on the two sides 
of the surface of discontinuity. 

For given p±, V\, equation (82.9) or (82.10) gives the relation between p2 
and V 2 . This relation is called the shock adiabatic or the Hugoniot adiabatic 
(W. J. M. Rankine, 1870; H. Hugoniot, 1889). It is represented graphically 
in the^F-plane (Fig. 43) by a curve passing through the given point (pi, V{) 



f Here we write the positive square root, since, as we shall see later (§84) we must have v x — v 2 > 0. 



482 



The shock adiabatic 



321 



(for pi = p2, Vi = V 2 we have also ei = e 2 , so that (82.10) is satisfied identi- 
cally). It should be noted that the shock adiabatic cannot intersect the vertical 
line V = V\ except at (pi, Vi). For the existence of another intersection 
would mean that two different pressures satisfying (82.10) correspond to the 
same volume. For V\ - V 2 , however, we have from (82.10) also ei = e 2 , 
and when the volumes and energies are the same the pressures must be the 
same. Thus the line V = V\ divides the shock adiabatic into two parts, 
each of which lies entirely on one side of the line. Similarly, the shock 
adiabatic meets the horizontal line p = pi only at the point {pi, Vi). 



ba 




Fig. 44 



Let aa' (Fig. 44) be the shock adiabatic through the point (pi, Vi) as a 
state of gas 1. We take any point (p2, V 2 ) on it and draw through that point 
another adiabatic bb', for which (p 2 , V 2 ) is a state of gas 1. It is evident that 
the pair of values (pi, Vi) satisfies the equation of this adiabatic also. The 
adiabatics aa' and bb' therefore intersect at the two points (pi, Vi) and (p 2 , 
V 2 ). It must be emphasised that the adiabatics are not identical, as would 
happen for Poisson adiabatics through a given point. This is a consequence 
of the fact that the equation of the shock adiabatic cannot be written in the 
form /(p, V) = constant, where / is some function, whereas the Poisson 
adiabatic, for example, can be written s(p, V) — constant. The Poisson 
adiabatics for a given gas form a one-parameter family of curves, but the 
shock adiabatic is determined by two parameters, the initial values pi and 
V\. This has also the following important result: if two (or more) successive 
shock waves take a gas from state 1 to state 2 and from there to state 3, the 
transition from state 1 to state 3 cannot in general be effected by the passage 
of any one shock wave. 

For a given initial thermodynamic state of the gas (i.e. for given pi and 
Vi), the shock wave is defined by only one parameter; for instance, if the 
pressure p 2 behind the shock is given, then Vi is determined by the Hugoniot 
adiabatic, and the flux density; and the velocities v\ and v 2 are then given by 
formulae (82.4) and (82.6). It should be mentioned, however, that we are 



322 Shock Waves §83 

here considering the shock wave in a co-ordinate system in which the gas is 
moving normal to the surface. If the shock wave may be situated obliquely 
to the direction of flow, another parameter is needed; for example, the value 
of the velocity component tangential to the surface. 

The following convenient graphical interpretation of formula (82.6) 
may be mentioned. If the point (p ly Vi) on the shock adiabatic (Fig. 43) 
is joined by a chord to any other point (p 2 , V 2 ) on it, then (p 2 -pi)l(V 2 -Vi) 
= -j 2 is Just the slope of this chord relative to the axis of abscissae. Thus;', 
and therefore the velocity of the shock wave, are determined at each point 
of the shock adiabatic by the slope of the chord joining that point to the 
point (pi,; Fi). 

Like the other thermodynamic quantities, the entropy is discontinuous at 
a shock wave. By the law of increase of entropy, the entropy of a gas can 
only increase during its motion. Hence the entropy s 2 of the gas which has 
passed through the shock wave must exceed its initial entropy s± : 

*2 > si. (82.11) 

We shall see below that this condition places very important restrictions on 
the manner of variation of all quantities in a shock wave. 

The following fact should be emphasised. The presence of shock waves 
results in an increase in entropy in those flows which can be regarded as 
motions of an ideal fluid in all space, the viscosity and thermal conductivity 
being zero. The increase in entropy signifies that the motion is irreversible, 
i.e. energy is dissipated. Thus the discontinuities are a means by which 
energy can be dissipated in the motion of an ideal fluid. It follows that 
d'Alembert's paradox (§11) does not arise when bodies move in an ideal fluid 
in such a way as to cause shock waves. In such cases there is a drag force. 

The true mechanism by which the entropy increases in shock waves lies, 
of course, in dissipative processes occurring in the very thin layers which 
actual shock waves are (see §87). It should be noticed, however, that the 
amount of this dissipation is entirely determined by the laws of conservation 
of mass, energy and momentum, when they are applied to the two sides of 
such layers; the width of the layers is just such as to give the increase in en- 
tropy required by these conservation laws. 

The increase in entropy in a shock wave has another important effect on 
the motion : even if we have potential flow in front of the shock wave, the flow 
behind it is in general rotational. We shall return to this matter in §106. 

§83. Weak shock waves 

Let us consider a shock wave in which the discontinuity in every quantity 
is small; we call this a weak shock wave. We transform the relation (82.9) 
by expanding in powers of the small differences *2-*i and p 2 -pi. We 
shall see that the first- and second-order terms in p 2 —pi then cancel; we 
must therefore carry the expansion with respect top 2 -pi as far as the third 



§83 Weak shock waves 323 

order. In the expansion with respect to $2 - *i, only the first-order terms need 
be retained. We have 

W2-W1 = (dw/dsi)p(s2-si) + (dwl8pi) s (p2-pi) + 

+ ftdholdpflfa -pif + U^l d Pi 3 )s{p2 -pi) 3 . 

By the thermodynamic identity dw = T ds + V dp we have for the derivatives 

(dzv/ds)p = T, {dwldp) s = V. 

Hence 

W2 — »i = Ti(s2-si)+Vi(p2-pi) + 

+ wvidpdfa -pi) 2 + \{&v\dpf) s (p>2 -pif. 

The volume Vz need be expanded only with respect top2—pi, since the second 
term of equation (82.9) already contains the small difference p2 —pi, and an 
expansion with respect to $2 — si would give a term of the form ($2 — si)(p2 —pi)> 
which is of no interest. Thus 

Vt-Vi = (dVldpdsipz-pj+^V/dp^fo-piY. 

Substituting this expansion in (82.9), we obtain 

1 I dW \ 

^-mm)*-**- (83,1) 

Thus the discontinuity of entropy in a weak shock wave is of the third order 
of smallness relative to the discontinuity of pressure. 

In all cases that have been investigated, the compressibility —(dVjdp) s 
decreases with increasing pressure, i.e. the second derivative 

(dW\dp*) s > 0. (83.2) 

It should be emphasised, however, that this is not a thermodynamic relation, 
and cannot be derived by thermodynamic arguments. It is therefore possible 
in principle that the derivative might be negative. We shall find several times 
in what follows that the sign of the derivative (d 2 VI8p 2 ) s is very important 
in gas dynamics. In future we shall assume it to be positive-! 

Let us draw through the point 1 (pi, Vi) in the pV-plane two curves, the 
shock adiabatic and the Poisson adiabatic. The equation of the latter is 
s 2 — si = 0. By comparing this with the equation (83.1) of the shock adiabatic 
near the point 1, we see that the two curves have contact of the second order at 
this point, both the first and the second derivatives being equal. In order to 
decide the relative position of the two curves near the point 1, we use the 
fact that, according to (83.1) and (83.2), we must have s 2 > si on the shock 
adiabatic for p2 > pi, while on the Poisson adiabatic S2 = *h The abscissa 
of a point on the shock adiabatic must therefore exceed that of a point on 



t For a perfect gas (0*F/fip*)g = (y+l)^/v 2 / >2 * This expression can be most simply obtained by 
differentiating Poisson's adiabatic equation pVY = constant. 



324 



Shock Waves 



§83 



tfye Poisson adiabatic having the same ordinate p^. This follows at once 
from the fact that, by the well-known thermodynamic formula (dV\ds) v 
— {Tjc v )(dVjdT) v , the entropy increases with the volume at constant 
pressure for all substances which expand on heating, i.e. which have {dV\dT) v 
positive. We can similarly deduce that, for p2 < pi, the abscissa of a point 
on the Poisson adiabatic exceeds that of the corresponding point on the shock 
adiabatic. Thus, near the point of contact, the two curves lie as shown in 
Fig. 45 (HH' being the shock adiabatic and PP' the Poisson adiabatic)f, 
both being concave upwards, by (83.2). 




Fig. 45 



For small p2-pi and V2-V1, formula (82.6) can be written, in the first 
approximation, as j 2 = —{dpjdV) s (we take the derivative for constant 
entropy, since the tangents to the two adiabatics at the point 1 coincide). 
The velocities v\ and V2 are, in the same approximation, equal : 

Vl = v 2 = v = jV = V[- V2(dp/dV) s ] = V( 8 P/dp)s. 

This is just the velocity of sound c. Thus the rate of propagation of weak 
shock waves is, in the first approximation, the velocity of sound : 

v = c. (83.3) 

From the properties of the shock adiabatic near the point 1 derived above 
we can deduce a number of important consequences. Since we must have 
$2 > $i in a shock wave, it follows Xh.2A.p2 > pi, i.e. the point 2 (j>2, V2) must 
lie above the point 1. Moreover, since the chord 12 has a greater slope than 
the tangent to the adiabatic at the point 1 (Fig. 43), and the slope of the 
tangent is equal to the derivative (dp/dVi) Sl , we have p > -(dpjdVi) Sl . 
Multiplying both sides of this inequality by Vi 2 , we find 

j2 Vl 2 = Vl 2 > _ VftdpldVi)* = (dpjd pi ) Sl = Cl 2 , 

where c\ is the velocity of sound corresponding to the point 1. Thus v\ > c\. 



t If (dVjdT)p is negative, the relative position is reversed. 



§84 The direction of "variation of quantities in a shock wave 325 

Finally, from the fact that the chord 12 has a smaller slope than the tangent 
at the point 2, it follows in like manner that v 2 < c 2 .-\ 

§84. The direction of variation of quantities in a shock wave 

The results of §83 show that, if the derivative (d 2 Vldp 2 ) s is assumed 
positive, it can be demonstrated very simply that for weak shocks the con- 
dition of increasing entropy (s 2 > $i) necessarily means that 

p 2 > pi, (84.1) 

vi > c\, v 2 < C2. (84.2) 

From the remark made concerning (82.6) it follows that, if p2 > pi, then 

Vi > V 2 , (84.3) 

and, since v±jVi = v 2 jV 2 = j, also 

©i > v 2 . (84.4) 

We shall now show that all these inequalities actually hold (still on the 
assumption that (d 2 Vldp 2 ) s is positive) for shock waves of any intensity. 
We shall therefore conclude, in particular, that, when gas passes through a 
shock wave, it is compressed, the pressure and density increasing (E. Jouguet, 
1904; G. ZemplEn, 1905)4 This means, graphically, that only the upper 
branch of the shock adiabatic (above the point 1) has any real significance; 
shock waves corresponding to points on the lower branch cannot exist. We 
may also mention the following important result which can be derived from 
the inequalities (84.2). Since a shock wave moves relative to the gas in front 
of it with a velocity vi > c\, it is clear that no perturbation starting from 
the shock wave can penetrate into that gas. In other words, the presence of 
the shock has no effect on the state of the gas in front of it. 

We shall now prove these statements, beginning with a preliminary 
calculation. We differentiate the relations (82.5) and (82.8) with respect 
to the quantities pertaining to gas 2, assuming the state of gas 1 to be un- 
changed. This means that pi, V± and wi are regarded as constants, while 
pi, V 2 , W2 and also^ (which depends on £2 and V 2 ) are differentiated. From 
(82.5) we obtain 

Fid(j2) = dp 2 +j*dV 2 +V 2 d(P), 



or 



dp 2 +j 2 dV 2 = ( Vi - V 2 )d(j 2 ), (84.5) 



t It can easily be shown in the same way that, when the derivative (8 2 V/dp 2 )s is negative, the con- 
dition s 2 > j x for weak shock waves implies that p 2 < /> x , while the velocities again satisfy v t > c lt 
v 2 < c 2 . 

{ If we change to a co-ordinate system in which gas 1 (in front of the shock wave) is at rest, and 
the shock is moving, then the inequality v t > v 2 means that the gas behind the shock wave moves 
(with velocity v t —v 2 ) in the same direction as the shock itself. 



326 Shock Waves 

and from (82.8) 

dw 2 +p V 2 dV 2 = Wi 2 - V 2 2 )d{j 2 ) y 
or, expanding the differential dw 2 , 

T 2 ds 2 + V 2 dp 2 +j 2 V 2 dV 2 = i(V^- V 2 2 )d(j 2 ). 
Substituting this equation in (84.5), we obtain 

T 2 ds 2 = i(V 1 -V 2 fd(P). 
Hence we see that 

d(j2)/d* 2 > 0, 

i.e. j 2 increases with s 2 . 



§84 



(84.6) 
(84.7) 



> 




Fig. 46 



We now show that there can be no point on the shock adiabatic at which it 
touches any line drawn from the point 1 (such as the point O is in Fig. 46). 
At such a point the slope of the chord from the point 1 is a minimum, and/ 2 
has a corresponding maximum, so that d(j 2 )jdp 2 = 0. We see from (84.6) 
that in this case we also have ds 2 Jdp 2 = 0. Next, substituting in (84.5) the 
differential dV 2 in the form dV 2 = (dV 2 fdp 2 ) Si dp 2 + (dV 2 lds 2 ) P2 ds 2 and ds 2 
in the form given by (84.6), and dividing by dp 2 , we obtain 

\ fy2 Is, J \ T 2 \ 8s 2 ) J dp 2 

Hence it follows that, for d(j 2 )/dp2 = 0, we must have 

Wf{W 2 \dp 2 ) Sl = \-v 2 2 \c 2 2 = 0, 

i.e. v 2 — c%\ conversely, if v 2 = c 2 , it follows that d(J 2 )jdp 2 = O.f 
Thus, of the three equations 

d(p)fdp2 = 0, ds 2 /dp 2 = 0, v 2 = c 2 , (84.8) 

each implies the other two and all three would hold at the point O (Fig. 46). 



t The expression in the braces can vanish only by chance, and this possibility is therefore 
unlikely. 



§84 The direction of variation of quantities in a shock wave 327 

Finally, we have for the derivative of j 2 (dV2/ dp2)s 2 = — z>2 2 /f2 2 at the point O 

dp 2 \ c 2 2 J J \ dp 2 */ S2 ' 

On account of the assumption that (d 2 Vjdp 2 ) s is positive, we therefore have 
at O 

d(v2lc 2 )ldp 2 < 0. (84.9) 

It is now easy to show that such a point cannot exist on the shock adiabatic. 
At points just above the point 1, ^2/^2 < 1 (see the end of §83). The equation 
V2JC2 = 1 can therefore be satisfied only by an increase in ^2/^2 ; that is, at O 
we should necessarily have d(v2/c2)ldp2 > 0, whereas by (84.9) the converse 
is true. In an entirely similar manner, we can show that the ratio ^2/^2 also 
cannot become equal to unity on the part of the shock adiabatic below the 
point 1. 

From the impossibility of the existence of a point such as O, which has 
just been demonstrated, we can at once deduce from the graph of the shock 
adiabatic that the slope of the chord from the point 1 (/>i, Vi) to the point 2 
(p2, V2) decreases as we move up the curve, and/ 2 correspondingly increases. 
From this property of the shock adiabatic and the inequality (84.7) it follows 
immediately that the necessary condition S2 > si implies that P2 > P\ also. 

It is also easy to see that, on the upper part of the shock adiabatic, the 
inequalities V2 < C2, v\ > c\ hold. The former follows at once from the fact 
that it holds near the point 1, and the ratio ^2/^2 can never become equal to 
unity. The second inequality follows from the fact that every chord from the 
point 1 to a point 2 above it is steeper than the tangent to the adiabatic at the 
point 1, since the curve cannot behave as shown in Fig. 46. 

The condition $2 > s\ and all three inequalities (84.1), (84.2) are therefore 
satisfied on the upper part of the shock adiabatic. On the lower part, however, 
none of these conditions holds. They are consequently equivalent, and if one 
is satisfied so are all the others. 

In the preceding discussion we have everywhere assumed that the deriva- 
tive (d 2 Vjdp 2 ) s is positive. If this derivative could change sign, it would no 
longer be possible to draw from the necessity of $2 > *i any general conclu- 
sions concerning inequalities for the other quantities. It is important, 
however, that the inequalities (84.2) for the velocities can be obtained by 
quite different arguments, which show that shock waves in which those 
inequalities do not hold cannot exist, even if their existence would not 
be disproved by the purely thermodynamic arguments given above. 

The reason is that we have still to discuss the subject of he stability of shock 
waves. Let us suppose that a shock wave at rest is subjected to an infinite- 
simal displacement in a direction (say) perpendicular to its plane. It can be 
shown that the result of such a displacement is that the shock wave is con- 
tinually accelerated in some direction, and it is clear that this demonstrates 
the absolute instability of such a wave and the impossibility of its existence. 



328 



Shock Waves 



§84 



The displacement of the shock wave is accompanied by infinitesimal 
perturbations in the gas pressure, velocity, etc. on both sides of the surface 
of discontinuity. These perturbations near the shock are then propagated 
away from it with the velocity of sound (relative to the gas) ; this, however, 
does not apply to the perturbation in the entropy, which is transmitted only 
with the gas itself. Thus an arbitrary perturbation of the type in question can 
be regarded as consisting of sound disturbances propagated in gases 1 and 2 
on both sides of the shock wave, and a perturbation of the entropy; the latter, 
which moves with the gas, will evidently occur only in gas 2 behind the shock. 
In each of the sound disturbances, the changes in the various quantities are 
related by certain formulae which follow from the equations of motion (as 
in any sound wave, §63), and therefore any such disturbance is specified 
by only one parameter. 



1'1>C, / 2 <C 2 



"l< C l V Z< C 2 



Fig. 47 



Let us now compute the number of possible sound disturbances. It 
depends on the relative magnitudes of the gas velocities v\, vi and the sound 
velocities c\, c^. We take the direction of motion of the gas (from 1 to 2) 
as the positive direction of the x-axis. The rate of propagation of the distur- 
bance in gas 1 relative to the stationary shock wave is u± = v±±ci, and in 
gas 2 it is 14,2 = ^2 ± £2- Since these disturbances must be propagated away from 
the shock wave, it follows that u\ < 0, m > 0. 

Let us suppose that v\> c\, vi < c^. Then it is clear that both values 
u\ = v\± c\ are positive, while only v% + c% of the two values of ui is positive. 
This means that the sound disturbances in which we are interested cannot 
exist in gas 1, while in gas 2 there can be only one, which is propagated relative 
to the gas with velocity c^. The calculation in other cases is similar. 

The result is shown in Fig. 47, where each arrow corresponds to one sound 
disturbance, propagated relative to the gas in the direction shown by the 
arrow. Each sound disturbance is defined, as stated above, by one parameter. 
Furthermore, in all four cases there are two other parameters, one determining 
the entropy perturbation propagated in gas 2 and one determining the dis- 
placement of the shock wave. For each of the four cases in Fig. 47, the 



§85 Shock waves in a perfect gas 329 

number in a circle shows the total number of parameters, thus obtained, 
which define an arbitrary perturbation arising from the displacement of the 
shock wave. 

The number of boundary conditions which must be satisfied by a pertur- 
bation on the surface of discontinuity is three (the continuity of the mass, 
energy and momentum fluxes). The solution of the stability problem is 
effected by prescribing the displacement of the shock wave (and therefore the 
perturbations in all the other quantities) in a form proportional to e nt , and 
determining the possible values of Q by means of the boundary conditions; 
the existence of real positive values of Q indicates absolute instability. In 
all except the first of the cases shown in Fig. 47, the number of parameters 
available exceeds the number of equations given by the boundary conditions 
at the discontinuity. In these cases, therefore, the boundary conditions 
admit any (and therefore any positive) value of O, and the shock wave is 
absolutely unstable. In the one case v\ > c\, V2 < c%, however, the number of 
parameters just equals the number of equations, and these therefore give a 
definite value of Q.. It is evident, without writing down the equations, that 
this value must be Q, = 0, since the problem contains no parameter of the 
dimensions sec -1 which could determine a value of Q. different from zero but 
not arbitrary. There is therefore no such instability in this case. 

Thus we see that the inequalities (84.2) for the velocity of the shock 
wave are necessary for the shock to exist, whatever the thermodynamic 
properties of the gas. 

In order to decide the stability of shock waves for which the condition 
(84.2) is satisfied, we should have to investigate also the other possible modes 
of instability. One of these is instability with respect to perturbations of the 
kind considered in §30 (characterised by periodicity in the direction parallel 
to the surface of discontinuity and forming "ripples" on this surface). We 
shall not perform the calculations here, but merely mention that shock waves 
are almost always stable with respect to such perturbations. Instability can 
occur only for certain very special forms of the shock adiabatic, which seem 
hardly ever to occur in Nature; they all require that the derivative (d 2 Vjdp 2 ) 8 
should be of variable sign. j- 

A shock wave might also, in principle, be unstable with respect to break- 
up into more than one surface of discontinuity. This problem has not been 
adequately investigated, but such instabilities may likewise occur only for 
certain very special types of shock adiabatic. 

§85. Shock waves in a perfect gas 

Let us apply the general relations obtained in the previous sections to 
shock waves in a perfect gas. The heat function of a perfect gas is given by 
the simple formula w = ypVI(y—l). Substituting this expression in (82.9), 



■f See S. P. D'yakov, Zhurnal eksperimental'noi i teoreticheskoi fiziki 27, 288, 1954; V. M. 
Kontorovich, ibid. 33, 1525, 1957; Soviet Physics JETP 6 (33), 1179, 1958. 



330 



Shock Waves 



§85 



we have after a simple transformation 



Yl_ = (r+l)j>i + (y-l)j>2 

V ± (y-l)p 1 + (y+l)p 2 ' 



(85.1) 



Using this formula, we can determine any of the quantities p h V ly p 2 , V 2 
from the other three. The ratio V 2 \V\ is a monotonically decreasing function 
of the ratio p 2 /p h tending to the finite limit (y - l)/(y + 1). The curve showing 
p 2 as a function of V 2 for given p h V\ (the shock adiabatic) is represented in 
Fig. 48. It is a rectangular hyperbola with asymptotes V 2 \V\ = (y- l)/(y + 1), 
pz/pi = - (y- l)/(y + 1). As we know, only the upper part of the curve, above 
the point V 2 jVi = p 2 \p\ — 1, has any real significance; it is shown in Fig. 48 
(for y — 1 -4) by a continuous line. 




For the ratio of the temperatures on the two sides of the discontinuity 
we find, from the equation of state for a perfect gas T 2 \T\ = p 2 V 2 \p\V\ y 
that 



72 

7\ 



Pi (y+l)/>i + (y-l)/>2 



(85.2) 



p! (y-l)pi + (y+l)p 2 
For the flux density; we obtain from (82.6) and (85.1) 

i 2 = {(y- l)/>i + (y + l)fr}/2Fi, (85.3) 

and then for the velocities of propagation of the shock wave relative to the 
gas before and behind it 

^i 2 = ^i{(y-i)/>i+(y+l)M 

*2 2 = Wi(v+ l)/»i + (y- lW 2 /{(y- l)/>i+(y+ l)/> 2 }. 



(85.4) 



§85 Shock waves in a perfect gas 331 

We may derive limiting results for very strong shock waves, in which p% 
is very large compared with^i.f From (85.1) and (85.2) we have 

v 2 lv 1 = P1 i P 2 = (y-i)i( y +i), ra/ri = (y-iMy+i)fr- ( 85 - 5 ) 

The ratio T%\T\ increases to infinity with pzjpi, i.e. the temperature discon- 
tinuity in a shock wave, like the pressure discontinuity, can be arbitrarily 
great. The density ratio, however, tends to a constant limit; e.g., for a 
monatomic gas the limit is pi — 4/>i, and for a diatomic gas p2 = 6pi. The 
velocities of propagation of a strong shock wave are 

oi = VWy+ifaVi}* V* = V(i(y- i)W/(y + 1»- (85.6) 

They increase as the square root of the pressure p^. 

Finally, we may give some formulae useful in applications, which express 
the ratios of densities, pressures and temperatures in a shock wave in terms of 
the Mach number Mi = v\\c\. These formulae are easily derived from the 
foregoing results : 

P2IP1 = t*/t* = (y+ l)Mi*/{(y- l)Mi* + 2}, (85.7) 

p 2 / Pl = 2yMi2/(y+ 1)- (y - l)/(y + 1), (85.8) 

Ta/Ti = {2yM 1 2-( y - l)}{(y- l)Mi2 + 2}/(y+ l)2Mi. (85.9) 
The Mach number M2 is given in terms of Mi by 

M 2 2 = (2 + (y~l)Mi2}/{2yMi2-(y-l)}. (85.10) 

PROBLEMS 

Problem 1. Derive the formula v x v 2 = c* 2 , where c* is the critical velocity. 

Solution. Since to+^v 2 is continuous at a shock wave, we can define a critical velocity 
which is the same for gases 1 and 2 by 

yp 1 .12 y P 2 ,i2 r+1 2 



* > 



(y-l)pi (y~l)p2 2(y-l) 

cf. (80.7). Determining p 2 lp 2 and Pifpi from these equations and substituting in 

p2 p\ 

Vl — V* = 

P2V2 piVl 

(obtained by combining (82.1) and (82.2)), we obtain 

y+1 / r.2 



-(t>i-i*)(l- — ) =0. 



2y \ V1V2 - 

Since v t # v 2 , this gives the required relation. 

Problem 2. Determine the value of the ratio p 2 /Pi, for given temperatures T u T 2 at a dis- 
continuity in a perfect gas with a variable specific heat. 



t It is necessary that not only p 2 ^> p x but p 2 ^> (y+ l)/>i/(y — !)• 



332 Shock Waves §85 

Solution. In the general case of a perfect gas with variable specific heat, we can say only 
that u) (like e) is a function of temperature alone, and that p, Fand Tare related by the equa- 
tion of state pV = RTIvl. Solving equation (82.9) for pjp u we obtain 

Pi RT ' 2ft WU i?ft 2ft J + ft /' 

where Wi =* w(2 , 1 ), w 2 = w(T 2 ). 

Problem 3 . A plane sound wave meets normally a shock wave in a perfect gas. Determine the 
intensity of sound transmitted by the shock wave (D. I. Blokhintsev, 1945). f 

Solution. Since a shock wave is propagated with supersonic velocity relative to the gas 
in front of it, no sound wave can be reflected from it. In gas 2, behind the discontinuity, an 
ordinary lsentropic transmitted sound wave is propagated, and also a perturbation of the 
entropy (at constant pressure), which is propagated with the moving gas itself. 

We consider the process in a co-ordinate system in which the shock wave is at rest, and the 
gas moves through it in the positive direction of the a;-axis, the incident sound wave being 
propagated in this direction also. The perturbations on the two sides of the discontinuity are 
related by conditions obtained by varying the boundary conditions (82.1)-(82.3). As a result 
of the sound disturbance, the shock wave also begins to oscillate; denoting its oscillatory 
velocity by 8u, we must write the change in the velocities v u v 2 in the boundary conditions 
as 8vj_ — 8u, 8v 2 — 8u. Thus J 

ViSpi+pxfivi-Sll) = V 2 8p 2 +p2(8v2-8u), 

Spi+vi 2 8p 1 + 2p 1 v 1 (8v 1 -8u) = 8p2+v2 2 8p2 + 2p 2 v z {8v2-8u), 
8w 1 +v 1 (8vi-8u) = 8zv 2 +V2(8v2-8u). 

In the incident sound wave we have 

$Si = 0, S^i = (Ci/p{)8pi = 8pi/dpi, 8W! = 8pil P1 . 

The perturbation in medium 2 is composed of the sound wave and the "entropy wave", 
which we denote by one and two primes respectively: 

S* 2 ' =* 0, 8V = (C2/p2)8p2 = 8p 2 '/c 2 p2, 8w 2 = 8p2'[p 2 , 

8p2 r ' = 0, 8v 2 " = 0, 8zv 2 " = T28S2" = -C2*8 P2 "lp2{y-\) 

(for a perfect gas (dsjdp)^ — —cjp). 

These relations enable us to express all quantities in the transmitted waves in terms of the 
corresponding quantities in the incident wave. The ratio of pressures in the sound waves is 
found to be 

8p 2 ' Mi + 1 /2( y -l)M 1 M 2 2 (Mi2-l)-(Mi-f-l)[( y -l)Mi2 + 2] 



8p! M 2 + 1 I 2( r - l)M 2 2(Mi2 - 1) - (M 2 + l)[( y - l)Mi2 + 2] 

For a weak shock wave (p 2 —pi <^Pi) we find 

8P2 . y+1 p2~Pl 

« 1 H , 

8pl 2y />! 



f The solution of the more general problem of oblique incidence of sound on a shock wave in an 
arbitrary medium is given by V. M. Kontorovich, Zhurnal experimental' noi i teoreticheskoi fiziki 
33, 1527, 1957; Soviet Physics JETP 6 (33), 1180, 1958. 

J We here denote the variable parts of quantities by 8 instead of the usual prime. 



§86 



Oblique shock waves 



333 



and in the opposite limiting case of a strong shock wave 

8p2 1 p2 

SpT ~ y+V[2y(y-l)] pi 

In both cases the pressure amplitude in the transmitted wave is greater than that in the 
incident wave. 

§86. Oblique shock waves 

Let us consider a steady shock wave, and abandon the system of co- 
ordinates used hitherto, in which the gas velocity is perpendicular to the shock 
surface element considered. The streamlines can intersect the surface of 
such a shock wave at any angle,f and in doing so are "refracted" : the tan- 
gential component of the gas velocity is unchanged, while the normal com- 
ponent is, according to (84.4), diminished: v\ t = ®2t, ^i» > ^2». It is there- 
fore clear that the streamlines "approach" the shock wave as they pass through 
it (cf. Fig. 49). Thus the streamlines are always refracted in a definite direc- 
tion in passing through a shock wave. 




Fig. 49 



The motion behind a shock wave may be either subsonic or supersonic 
(only the normal velocity component need be less than the velocity of sound 
C2) ; the motion in front of it is necessarily supersonic. If the gas flow on 
both sides is supersonic, every disturbance must be propagated along the 
surface in the direction of the tangential component of the gas velocity. In 
this sense we can speak of the "direction" of a shock wave, and distinguish 
shock waves leaving and reaching any point (as we did for characteristics, the 
motion near which is always supersonic; see §79). If the motion behind the 
shock is subsonic, there is strictly no meaning in speaking of its "direction", 
since perturbations can be propagated in all directions on its surface. 

We shall derive a relation between the two components of the gas velocity 
after it has passed through an oblique shock wave, supposing that we have a 
perfect gas. We take the direction of the gas velocity vi in front of the shock 
as the #-axis; let <f> be the angle between the shock and the #-axis (Fig. 49). 



f The only restriction is that the normal velocity component v ln exceeds c x . 



334 



Shock Waves 



§86 



The continuity of the velocity component tangential to the shock means that 
vi cos <f> *= V2x cos <f> + V2y sin <f>, or 



tan<£ = (vi-V2 X )jvzy. 



(86.1) 



Next we use formula (85.7), in which v\ and v 2 denote the velocity components 
normal to the plane of the shock wave and must be replaced by vi sin (f> and 
V2x sin <f> — vzy cos <£, so that 



V2x sin^ — V2y cos<f> y—\ 



v\ sin <f> 



+ 



2d 2 



y+1 (y+ l)^i 2 sin 2 </> 



(86.2) 




Fig. 50 



We can eliminate the angle § from these two relations. After some simple 
transformations, we obtain the following formula which determines the re- 
lation between v 2x and v 2y (for given vi and c{) : 



V 2 y 2 = (V 1 -V 2X ) 2 



2(V 1 -C 1 *lv 1 )l(y +l )-fo i - V 2x ) 
Vi - V2 X + 2ci 2 /(y + 1 )V! 



(86.3) 



This formula can be more intelligibly written by introducing the critical 
velocity. According to Bernoulli's equation and the definition of the critical 
velocity, we have m + W = ^i 2 + ci 2 /(y-l) = (y+l>v72( y -l) (see §85, 
Problem 1), whence 



c**= [(y-l)^ 2 + 2 tl 2 ]/( r +l). 
Using this in (86.3), we obtain 



V 2 y 2 = (Vl-V2x) 2 ~ 



VlV2x-C* 2 



(86.4) 



(86.5) 



2vi 2 /(y + 1 ) - v±V2x + C* 2 

Equation (86.5) is called the equation of the shock polar. Fig. 50 shows a 
praph of the function V2y{v2x)\ it is a cubic curve, called a strophoid. It 
crosses the axis of abscissae at the points P and Q, corresponding to V2x 



§86 Oblique shock waves 335 

= c^\v\ and v% x = »i«t A li ne (OS in Fig. 50) drawn from the origin at an 
angle x to the axis of abscissae gives, by the length of the segment between O 
and the point where it intersects the shock polar, the gas velocity behind a 
discontinuity which turns the stream through an angle x- There are two such 
intersections (A, B), i.e. two different shock waves correspond to a given 
value of X- The direction of the shock wave also can be immediately deter- 
mined from the shock polar: it is given by the direction of the perpendicular 
from the origin to the line QB or QA (Fig. 50 shows the angle for a shock 
corresponding to the point B). As % decreases, the point A approaches P, 
corresponding to a normal shock {<f> = \tt) with V2 = c^-\v\. The point B 
approaches Q\ the intensity of the shock (velocity discontinuity) tends 
to zero, and the angle cf> tends, as it should, to the Mach angle a = 
sin -1 (cijvi)\ the tangent to the shock polar at Q makes an angle £77+ a with 
the axis of abscissae. 

From the shock polar we can immediately derive the important result that 
the angle of deviation x of the stream at the shock wave cannot exceed a certain 
maximum value xmax, corresponding to the tangent from O to the curve. 
This quantity is, of course, a function of the Mach number Mi = vijci, 
but we shall not give the expression for it, which is very cumbersome. For 
Mi = 1, xmax = 0; as Mi increases, xmax increases monotonically, and tends 
to a finite limit as Mi ->• 00. It is easy to discuss the two limiting cases. 
If the velocity v\ is near to c+, then V2 is so also, and the angle x is small; 
the equation (86.5) of the shock polar can then be written in the approximate 
formj 

x 2 = (y+l)(v 1 -v 2 ) 2 (v 1 +V2-2c m )l2cJ, (86.6) 

where we have put v% x ~ ^2, V2y ~ c*x i n view of the smallness of x- Hence 
we easily findff 

W(y +l) ("i ,\« V2 , „ n , , s , 7 , 

^--WJ-W.- 1 ) = 3V3(y+D (Ml-1) - (86 ' 7) 

In the opposite limiting case Mi = 00 (i.e. Mi* = V[(y+l)/(7 — l)])> tne 
shock polar degenerates to a circle which meets the axis of abscissae at the 
points c^[(y— l)/(y + 1)] and c^y/fty + l)/(y — 1)]. It is easy to see that we 
then have 

Xmax = sin-i(l/ y ); (86.8) 



f The strophoid actually continues in two branches from the point v 2 x = v x (which is a double 
point) to infinite f 2 j/> these are not shown in Fig. 50. They have a common asymptote 

vzx = c* 2 lvi + 2vi((y+l). 

The points on these branches have no physical significance; they would give values for v 2 x and V2y 
such that v 2 Jv ln > 1, which is impossible. 

J It is easily seen that equation (86.6) holds also for any (non-perfect) gas, provided that (y+1) 
is replaced by 2a* (95.2). 

If It may be noted that this dependence of Xmax on M x — 1 is in agreement with the general simi- 
larity law (118.7) for transonic flow. 



336 



Shock Waves 



§86 



for air this is 45-6°. Figure 51 shows a graph of xmax as a function of Mi for 
air; the Upper curve is a similar graph for flow past a cone (see §105). 

The circle v% = c* cuts the axis of abscissae between the points P and 
Q (Fig- 50), and therefore divides the shock polar into two parts correspond- 
ing to subsonic and supersonic gas velocities behind the discontinuity. The 
point where this circle crosses the polar lies to the right of, but very close to, 
the point C; the whole segment PC therefore corresponds to transitions to 
subsonic velocities, while CQ (except for a very small segment near C) 
corresponds to transitions to supersonic velocities. 




Fig. 51 



For given Mi and </>, the pressure change in the shock wave is given by 
p 2 2yMi 2 sin 2 <i-(y-l) 

T = 1 /n ; < 86 - 9 ) 

Pi (y+i) 

this is formula (85.8) with Misin<£ in place of Mi. This ratio increases 
monotonically when the angle <f> increases from its smallest value sin -1 (l/Mi) 
(when P2IP1 = 1) to \n, i.e. as we move along the shock polar from Q to P. 
The two shock waves determined by the shock polar for a given deviation 
angle x are often said to belong to the weak and strong families. A shock 
wave of the strong family (the segment PC of the polar) is strong (the ratio 
P2IP1 is large), makes a large angle <f> with the direction of the velocity vi, 
and converts the flow from supersonic to subsonic. A shock wave of the weak 
family (the segment QC) is weak, is inclined at a smaller angle to the stream, 
and almost always leaves the flow supersonic. 

PROBLEMS 

Problem 1 . Derive the formula giving the angle of deviation x of the velocity in an oblique 
shock wave (in a perfect gas) in terms of Mi = vjc^ and the angle ^ between the shock 
wave and the direction of the velocity v x (Fig. 49) : 



cotx = tan^ 



|.2(Mi2sin2«i-l) J 



2(M r 



§87 The thickness of shock waves 337 

Problem 2. Derive the formula giving the number M 2 = vjc 2 in terms of M x and <£: 

2 + (y-l)Mi 2 2Mi 2 cos 2 <£ 

M2 2 = , . - . — + 



2yMi 2 sin 2 <f> - (y - 1) 2 + (y - l)Mi 2 sin 2 <£ 

§87. The thickness of shock waves 

Hitherto we have regarded shock waves as geometrical surfaces of zero 
thickness. We shall now consider the structure of actual surfaces of dis- 
continuity, and we shall see that shock waves in which the discontinuities 
are small are in reality transition layers of finite thickness, the thickness 
diminishing as the magnitude of the discontinuities increases. If the dis- 
continuities are not small, the change occurs so sharply that the concept of 
thickness is meaningless. 

To determine the structure and thickness of the transition layer we must 
take account of the viscosity and thermal conductivity of the gas, which we 
have hitherto neglected. 

The relations (82.1)-(82.3) for a shock wave were obtained from the 
constancy of the fluxes of mass, momentum and energy. If we consider a 
surface of discontinuity as a layer of finite thickness, these conditions must be 
written, not as the equality of the quantities concerned on the two sides of 
the discontinuity, but as their constancy throughout the thickness of the 
layer. The first condition, (82.1), is unchanged: 

pv =j= constant. (87.1) 

In the other two conditions additional fluxes of momentum and energy, due 
to internal friction and thermal conduction, must be taken into account. 

The momentum flux density (in the x-direction) due to internal friction is 
given by the component — & ' xx of the viscosity stress tensor ; according to the 
general expression (15.3) for this tensor, we have a xx — ON + Qjdvjdx. 
The condition (82.2) then becomes 

p+pv 2 — (jft + Qdv/dx = constant. 

As in §82, we introduce the specific volume V in place of the velocity v = jV. 
Since j = constant, dvjdx = jdVjdx, so that 

P +j 2 V— (f 77 + £)/ dVjdx = constant. 

At great distances from the shock wave, the thermodynamic quantities are 
constants, i.e. they are independent of x; in particular, dVjdx = 0. We 
denote by a suffix 1 the values of quantities far in front of the shock wave. 
Then we can put the constant equal to Pi+j 2 Vi, obtaining 

p-pi+P(V- V^-^rj + QjdV/dx = 0. (87.2) 

Next, the energy flux density due to thermal conduction is — k dTjdx. That 
due to internal friction is —a' X iVi or, since the velocity is along the #-axis, 
— o'xxv = —(£rj+£)vdv/dx. Thus the condition (82.3) can be written 

pv{w-\-\v 2 ) — {%-q + Qv dvjdx— k dTjdx = constant. 



338 Shock Waves §87 

Again putting v = jV, we can obtain the final form 

v>+$pV*-j($r} + QVdV/6x-(Klj)dT/dx = wi + i/W- (87.3) 

We shall here consider shock waves in which all the discontinuities are 
small. Then all the differences V— Vi, p—pi, etc. between the values inside 
and outside the transition layer are also small. In (87.2) we expand V-V\ 
in powers oip—pi and s — si, taking the pressure and entropy as the indepen- 
dent variables. It is seen from the relations obtained below that 1/S (where 
S is the thickness of the discontinuity) is of the first order in p —pi, and the 
difference s — s± is of the second order.f Hence we can write, neglecting quan- 
tities of the third order, 

F-Fi = (dVldp) s {p-p 1 )+^8Wl8p^ s {p-p 1 f + (8V/8s) p (s-s 1 ). 
The values of all the coefficients are, of course, taken outside the transition 
layer (i.e. for p = p h s = s{). Substituting this expansion in (87.2), we 
obtain 

[1 + (dV/8p) s p](p - Pl ) + ij2^2 V/8p 2 )s (p _ pi) 2 + (8V/8s) p (s - S1 )P 

= (tn + QjdV/dx. 

The derivative dF/dx can be written 

dV _ / 8V \ dp / 8V\ ds 

dx \ dp J s dx \ 8s Jp dx 

Differentiation with respect to x increases the order of smallness by one, 
since 1/8 is of the first order; the derivative dp/dx is therefore of the second 
order, and ds/dx of the third order. The term in ds/dx can therefore be omit- 
ted. Thus the condition (87.2) becomes 

[l^Vl^splip-p^HPi^V/Sp^ip-p^ + idV/Ss^is-s^P 

= (ft + Z>){Wl8p) s (dp/dx) j. (87.4) 

Next, we multiply each term of (87.2) by \{V+ V\) and subtract from 
equation (87.3). The result is 

ito-t 0l )-\{p-p 1 yy+ Vj-iM-n+ZW- v 1 )^-- K - ^- =0. 

dx j ax 

The third term, which contains the product (V—V\) dVjdx, is of the third 
order, and may be omitted : 

(w-wi)-i(p-^)i)(F+ri)-(/c/;")dT/dx = 0. 

The first two terms are just the expression which we expanded in powers of 
p—pi smds—si in deriving formula (83.1). The first- and second- order terms 
in p —p\ in this expansion are zero, and we have as far as terms of the second 

t The total entropy discontinuity s 2 — s x is, as we have seen in §83, of the third order relative to 
the pressure discontinuity p 2 —pi> whereas s— Sj is of only the second order in p—p x . The reason is 
that, as we shall show below, the pressure in the transition layer varies monotonically from p x to p 2 , 
whereas the entropy does not vary monotonically; it has a maximum within the layer. 



§87 The thickness of shock waves 339 



order just T(s-s{). The derivative dTjdx can be written 

dT _ / dT\ dp l 8T\ ds ^ / dT \ dp 

dx \ dp J s dx \ ds / „ dx \ dp / s dx 



The result is 



""-^-jUlsr (87 - 5) 



Substituting this expression for s — si in (87.4), we find 

"■(-J),»- w " + [' t (f)/] < '- w 

-(-y(?l(f ).*;»©.)&■ <«"> 

The flux j is, in the first approximation, j = vjV « c/F (see (83.3)). 
This expression can be substituted on the right-hand side of (87.6), but it 
will not serve on the left-hand side; further terms have to be included 
in^ 2 . These terms could be obtained from (87.2), for instance. It is simpler, 
however, to argue as follows. At great distances on both sides of the surface 
of discontinuity, the right-hand side of (87.6) is zero, since dpjdx is zero. 
At such distances the pressure is p\ or />2- That is, we can say that the 
quadratic in p on the left of (87.6) has the zeros p\ and p2. By a well-known 
theorem of algebra, it can therefore be written as the product (p —pi)(p —p?) 
multiplied by the coefficient of p 2 , ^j 2 (d 2 Vldp 2 ) s . 

Thus we have the following differential equation for the function p(x) :f 

\ I dW\ V 3 { k I dT \ / 8V \ n \ dp 

From the thermodynamic formulae for derivatives, {dV\ds) v = (dTjdp) s ; it is 
easy to see that the coefficient of — dp/dx on the right-hand side of the above 
equation is 2V 2 a, where a is related to the sound-absorption coefficient y 
(77.6) by y = aco 2 . Thus 

dp 1 1 d 2 V\ 

s--4»^M.»-^- w - (87 - 7) 

Integration gives 

\V 2 a r dp 

x = — 



J (P-: 



(d*V/dp 2 ) s J (p- Pl )(p-p 2 ) 

4aV2 p -i(p 2+pi ) 

tanh -1 h constant. 

i(p2-pi)(d 2 V/dp 2 ) s &P2-P1) 



t In considering a weak shock wave we can regard the viscosity and the thermal conductivity as 
constants. 



340 Shock Waves §87 

Putting the constant equal to zero, we have 

P ~ KP2 +pi) = I(p2 -pi) tanh(*/S), (87.8) 

where 

3 = 8aV 2 /(p 2 -pi)(d 2 V/dp 2 ) s . (87.9) 

This gives the manner of variation of the pressure between the values p\ 
and pi which it takes at great distances on the two sides of the shock wave. 
The point x — corresponds to the median value of the pressure, i(j>i+P2)- 
For x ->; ±oo, the pressure tends asymptotically to p\ and p2. Almost 
the whole change from p\ to p2 occurs over a distance of the order of S, 
which may be called the thickness of the shock wave. We see that this is the 
less, the stronger the shock, i.e. the greater the pressure discontinuity. 

The variation of the entropy across the discontinuity is obtained from 
(87.5) and (87.8) 

k i 8T\ i dW \ 1 

J - Sl = T6^^(^) s (^i-)^-^^^P)- ( 87 - 10 > 

From this we see that the entropy does not vary monotonically, but has a 
maximum inside the shock, at x = 0. For x = ± oo this formula gives 
s = si in either case; this is because the total entropy change $2— si is of the 
third order inp^—pi (cf. (83.1)), whereas s — si is of the second order. 

Formula (87.8) is quantitatively valid only for sufficiently small differences 
p2—pi- We can, however, use (87.9) qualitatively to determine the order of 
magnitude of the thickness in cases where the difference p2~Pi is of the 
same order of magnitude as pi and p2 themselves. The velocity of sound in 
the gas is of the same order as the thermal velocity v of the molecules. The 
kinematic viscosity is, as we know from the kinetic theory of gases, v ~ h 
~ Ic, where / is the mean free path of the molecules. Hence a ~ Ijc 2 ; an 
estimate of the thermal-conduction term gives the same result. Finally, 
(d 2 Vjdp 2 ) s ~ Vjp 2 , and pV ~ c 2 . Using these relations in (87.9), we obtain 

8 ~/. (87.11) 

Thus the thickness of a strong shock is of the same order of magnitude as the 
mean free path of the gas molecules.f In macroscopic gas dynamics, however, 
where the gas is treated as a continuous medium, the mean free path must be 
taken as zero. It follows that the methods of gas dynamics cannot strictly be 
used alone to investigate the internal structure of strong shock waves. 

A considerable increase in the thickness of a shock wave may be caused by 
the presence in the gas of comparatively slow relaxation processes (slow 
chemical reactions, a slow energy transfer between different degrees of free- 
dom of the molecule, and so on). This topic has been discussed by Ya. B. 
Zel'dovich (1946). 



t A strong shock wave causes a considerable increase in temperature; / denotes the mean free 
path for some mean temperature of the gas in the shock. 



§87 



The thickness of shock waves 



341 



Let t be of the order of magnitude of the relaxation time. Both the 
initial and the final states of the gas must be states of complete equilibrium; 
it is therefore immediately clear that the total thickness of the shock wave will 
be of the order of tz>i, the distance traversed by the gas in the time t. It 
is also found that, if the shock strength is above a certain limit, its structure 
becomes more complex; this may be seen as follows. 



p 2 




Fig. 52 



In Fig. 52 the continuous curve shows the shock adiabatic drawn through a 
given initial point 1, on the assumption that the final states of the gas are 
states of complete equilibrium; the slope of the tangent at the point 1 gives 
the "equilibrium" velocity of sound, denoted in §78 by cq. The dashed curve 
shows the shock adiabatic through the same point 1, on the assumption that 
the relaxation processes are "frozen" and do not occur. The slope of the 
tangent to this curve at the point 1 gives the velocity of sound denoted in 
§78 by <v 

If the velocity of the shock wave is such that Co < v± < c^, the chord 
12 lies as shown in Fig. 52 (the lower chord). In this case we have a simple 
increase in the shock thickness, all intermediate states between the initial 
state 1 and the final state 2 being represented in the pV-plane by points on 
the segment 12.-)- 

If, however, v\> c^, the chord takes the position 11 '2'. No point lying 
between 1 and 1' corresponds to any actual state of the gas ; the first real point 
(after 1) is 1', which corresponds to a state in which the relaxation equili- 
brium is no different from that in state 1. The compression of the gas from 
state 1 to state V occurs discontinuously, and afterwards (over distances 
~ v\r) it is gradually compressed to the final state 2' 



t This follows from the fact that (neglecting ordinary viscosity and thermal conduction) all the 
states through which the gas passes satisfy the equations of conservation of mass, pv = j = constant, 
and of momentum, p-\-pV = constant (cf. the similar but more detailed discussion in §121). 

12 



342 Shock Waves §88 

§88. The isothermal discontinuity 

The discussion of the structure of a shock wave in §87 involves the assump- 
tion that the viscosity and thermal conductivity are of the same order of 
magnitude, as is usually the case. The case where x > v is a ls° possible, 
however. If the temperature is sufficiently high, additional heat is transferred 
by thermal radiation in equilibrium with the matter. Radiation has a much 
smaller effect on the viscosity (i.e. the momentum transfer), and so v may 
be small compared with x- We shall now see that this inequality leads to a 
very important difference in the structure of the shock wave. 

Neglecting terms in the viscosity, we can write equations (87.2) and (87.3), 
which determine the structure of the transition layer, as 

p+PV = p 1 +jW lf (88.1) 

J rp 

- = w+ lj2 V 2_ Wl _lj2 Vl 2. ( 88-2 ) 

j dx 

The right-hand side of (88.2) is zero only at the boundaries of the layer. 
Since the temperature behind the shock wave must be higher than that in 
front of it, it follows that we have 

dT/dx > (88.3) 

everywhere in the transition layer, i.e. the temperature increases monotoni- 
cally. 

All quantities in the layer are functions of a single variable, the co-ordinate 
x, and therefore are functions of one another. Differentiating (88.1) with 
respect to V, we obtain 



\ dTJvdV \8VJt 



The derivative (dpjdT) v is always positive in gases. The sign of the derivative 
dTfdV is therefore the reverse of that of the sum (dpjdV)T +j 2 - In state 1 we 
have j 2 < —(dpidVil)s (since ^i > c{), and, since the adiabatic compressibility 
is always less than the isothermal compressibility,; 2 > — {dp\jdVi)T. On side 
1, therefore, dT\]dV\ < 0. If this derivative remains negative everywhere in 
the transition layer, then, as the gas is compressed (V decreasing), the 
temperature increases monotonically, in accordance with (88.3), from side 1 
to side 2, In other words, we have a shock wave whose thickness is much 
increased by the high thermal conductivity (possibly to such an extent that 
even to call it a shock wave is mere convention). 
If, however, the shock is so strong that 

P < -(8p 2 ldV 2 )T, (88.4) 

then we have in state 2 d^/dF^ > 0, so that the function T{V) has a maxi- 
mum somewhere between V\ and Vi (Fig. 53). It is clear that the transition 



§88 



The isothermal discontinuity 



343 



from state 1 to state 2, with V changing continuously, then becomes im- 
possible, since the inequality (88.3) can not be satisfied everywhere. 

Consequently, we have the following pattern of transition from the initial 
state 1 to the final state 2. First comes a region where the gas is gradually 
compressed from the specific volume V\ to some V (the value for which 
T(V) = T 2 for the first time; see Fig. 53); the thickness of this region is 
determined by the thermal conductivity, and may be considerable. The 
compression from V to V 2 then occurs discontinuously, the temperature 
remaining constant at T 2 . This may be called an isothermal discontinuity. 




Fig. 53 



Let us determine the variation of the pressure and density in an isothermal 
discontinuity, assuming that we have a perfect gas. The condition of con- 
tinuity of momentum flux (88.1), applied to the two sides of the discontinuity, 
gives p'+jW = p2+j 2 V 2 . For a perfect gas V = RTj^p; since T = T 2 , 
we have 

, j 2 RT 2 pRT 2 

P' + — — = P2 + J ——. 
\xp fip 2 

This quadratic equation for p' has the solutions p' = p 2 (trivial) and 

p' =pRT 2 / H ,p2=pV 2 . (88.5) 

We can express;* 2 in the form (82.6), obtaining p' = (p 2 -pi)V 2 l(Vi- V 2 \ 
and, substituting V 2 fVi from (85.1), we have 

/»' = |[(r+i)/»i+(y-i)M (88.6) 

Since we must have p 2 > p', we find that an isothermal discontinuity occurs 
only when the ratio of the pressures p 2 and pi satisfies 

P2/P1 > (y+l)/(3-y) (88.7) 

(Rayleigh 1910). This condition can, of course, be obtained directly from 
(88.4). 



344 Shock Waves §89 

Since, for a given temperature, the gas density is proportional to the 
pressure, the density ratio in an isothermal discontinuity is equal to the 
pressure ratio; 

P'lf* = V2IV = p'lpz. (88.8) 



§89. Weak discontinuities 

Besides surface discontinuities, at which the quantities p, p, v etc. are 
discontinuous, we can also have surfaces at which these quantities, though 
remaining continuous, are not regular functions of the co-ordinates. The 
irregularity may be of various kinds. For example, the first spatial derivatives 
of p, p, v etc. may be discontinuous on a surface, or these derivatives may 
become infinite; or higher derivatives may behave in the same manner. 
We call such surfaces weak discontinuities, in contrast to the strong discon- 
tinuities (shock waves and tangential discontinuities), in which the quantities 
p, p, v, ... themselves are discontinuous. 

It is easy to see from simple considerations that weak discontinuities are 
propagated relative to the gas (on either side of the surface) with the velocity 
of sound. For, since the functions />, p, v, ... themselves are continuous, 
they can be "smoothed" by modifying them only near the surface of dis- 
continuity, and only by arbitrarily small amounts, in such a way that the 
smoothed functions have no singularity. The true distribution of the 
pressure, say, can thus be represented as a superposition of a perfectly smooth 
function po, free from all singularities, and a very small perturbation p' 
of this distribution near the surface of discontinuity; and the latter, like any 
small perturbation, is propagated, relative to the gas, with the velocity of 
sound. 

It must be emphasised that, for a shock wave, the smoothed functions would 
differ from the true ones by quantities which in general are not small, and 
the foregoing arguments are therefore invalid. If, however, the discon- 
tinuities in the shock wave are sufficiently small, those arguments are again 
applicable, and such a shock wave is propagated with the velocity of sound, 
a result which was obtained by another method in §83. 

If the flow is steady in a given co-ordinate system, then the surface of dis- 
continuity is at rest in that system, and the gas flows through it. The gas 
velocity component normal to the surface must equal the velocity of sound. 
If we denote by a the angle between the direction of the gas velocity and the 
tangent plane to the surface, then v n = v sin a = c, or sin a = c\v, i.e. a 
surface of weak discontinuity intersects the streamlines at the Mach angle. 
In other words, a surface of weak discontinuity is one of the characteristic 
surfaces, a result which is entirely reasonable if we recall the physical signi- 
ficance of the latter : they are surfaces along which small perturbations are 
propagated (see §79). It is clear that, in steady flow of a gas, weak discon- 
tinuities can occur only at velocities not less than that of sound. 



§89 Weak discontinuities 345 

Weak discontinuities differ fundamentally from strong ones in the manner 
of their occurrence. We shall see that shock waves can be formed as a direct 
result of the gas flow, the boundary conditions being continuous (for instance, 
the formation of shock waves in a sound wave, §95). In contrast to this, weak 
discontinuities cannot occur spontaneously; they are always the result of 
some singularity of the initial or boundary conditions of the flow. These 
singularities may be of various kinds, like the weak discontinuities themselves. 
For example, a weak discontinuity may occur on account of the presence of 
angles on the surface of a body past which the flow takes place; in this case 
the first spatial derivatives of the velocity are discontinuous. A weak dis- 
continuity is also formed when the curvature of the surface of the body is 
discontinuous, without there being an angle; in this case the second spatial 
derivatives of the velocity are discontinuous, and so on. Finally, any sin- 
gularity in the time variation of the flow results in a non-steady weak dis- 
continuity. 

The gas velocity component tangential to the surface of a weak discon- 
tinuity is always directed away from the point (e.g. an angle on the surface 
of a body) from which the perturbation begins which causes the discontinuity; 
we shall say that the discontinuity begins from this point. This is an example 
of the fact that, in a supersonic flow, perturbations are propagated down- 
stream. 

The presence of viscosity and thermal conduction results in a finite 
thickness of a weak discontinuity, which is therefore in reality a transition 
layer, like a shock wave. The thickness of the latter, however, depends only 
on its strength and is constant in time, whereas the thickness of a weak 
discontinuity increases with time after its formation. It is easy to determine 
the law governing this increase. To do so, we again use the remark made at 
the beginning of this section, that the motion of any part of the surface of a 
weak discontinuity follows the same equations as the propagation of any weak 
perturbation in the gas. When viscosity and thermal conduction occur, a 
perturbation which is initially concentrated in a small volume (a "wave 
packet") expands as it moves in the course of time; the manner of this expan- 
sion has been determined in §77. We can therefore conclude that the thick- 
ness S of a weak discontinuity is of the order of 

8 ~ V(act), (89.1) 

where t is the time from the formation of the discontinuity and a the co- 
efficient of the squared frequency in the sound absorption coefficient. If 
the discontinuity is at rest, then the time t must be replaced by Ijc, where 
/ is the distance from the point where the discontinuity starts (e.g. for a weak 
discontinuity starting from an angle on the surface of a body, / is the distance 
from the vertex of the angle); consequently 8 ~ -\/(al). Thus the thickness 
of a weak discontinuity increases as the square root of the time from its 
formation or of the distance from its starting-point. 

To conclude this section, we should make the following remark, analogous 



346 Shock Waves §89 

to the one at the end of §79. We stated there that, among the various per- 
turbations of the state of a gas in motion, perturbations of entropy (at con- 
stant pressure) and vorticity are distinct in their properties. Such pertur- 
bations do not move relative to the gas, and are not propagated with the velocity 
of sound. Hence the surfaces at which the entropy and vorticityf are weakly 
discontinuous are at rest relative to the gas, and move with it relative to a 
fixed system of co-ordinates. Such discontinuities may be called weak 
tangential discontinuities; they pass through streamlines, and are in this respect 
entirely analogous to the strong tangential discontinuities. 



t A weak discontinuity of the vorticity implies a weak discontinuity of the velocity component 
tangential to the surface of discontinuity; for example, the normal derivatives of the velocity may 
be discontinuous. 



CHAPTER X 

ONE-DIMENSIONAL GAS FLOW 

§90. Flow of gas through a nozzle 

Let us consider steady flow of a gas out of a large vessel through a tube of 
variable cross-section (a nozzle). We shall suppose that the gas flow is 
uniform over the cross-section at every point in the tube, and that the velocity 
is along the axis of the tube. For this to be so, the tube must not be too wide, 
and its cross-sectional area S must vary fairly slowly along its length. Thus 
all quantities characterising the flow will be functions only of the co-ordinate 
along the axis of the tube. Under these conditions we can apply the relations 
obtained in §80, which are valid along streamlines, directly to the variation 
of quantities along the axis. 

The mass of gas passing through a cross-section of the tube in unit time 
(the discharge) is Q = pvS\ this must evidently be constant along the tube: 

Q = Spv = constant. (90.1) 

The linear dimensions of the vessel are supposed very large in comparison 
with the diameter of the tube. The velocity of the gas in the vessel may there- 
fore be taken as zero, and accordingly all quantities with the suffix in the 
formulae of §80 will be the values of those quantities in the vessel. 

We have seen that the flux density j = pv cannot exceed a certain limiting 
value j\. It is therefore clear that the possible values of the total discharge Q 
have (for a given tube and a given state of the gas in the vessel) an upper 
limit £ max , which is easily determined. If the value j m of the flux density 
were reached anywhere except at the narrowest point of the tube, we should 
have j > /„. for cross-sections with smaller S, which is impossible. The 
valued = jx can therefore be attained only at the narrowest point of the tube; 
let the cross-sectional area there be S min . Then the upper limit to the total 
discharge is 

£max = p.©.Snin = V(^0/>o)[2/(y+ l)]a+r)/2(y-l) /S ' mln . (QQ.2) 

Let us first consider a nozzle which narrows continually towards its outer 
end, so that the minimum cross-sectional area is at that end (Fig. 54). By 
(90.1), the flux density j increases monotonically along the tube. The same is 
true of the gas velocity v, and the pressure accordingly falls monotonically. 
The greatest possible value of j is reached if v attains the value c just at the 
outer end of the tube, i.e. if v\ — c\ = v* (the suffix 1 denotes quantities 
pertaining to the outer end). At the same time, p\= p*. 

Let us now follow the change in the manner of outflow of the gas when the 
external pressure p e diminishes. When this pressure decreases from po, the 

347 



348 



One-dimensional Gas Flow 



§90 



pressure inside the vessel, to />*, the pressure pi at the outer end of the tube 
decreases also, and the two pressures pi andp e remain equal; that is, the whole 
of the pressure drop from p to p e occurs in the nozzle. The velocity vi 
with which the gas leaves the tube, and the total discharge Q = j\S min , 
increase monotonically, however. For p e = p* the velocity becomes equal to 
the local velocity of sound, and the discharge reaches the value Q m&x . When 
the external pressure decreases further, the pressure pi remains constant 
at p*, and the fall of pressure from p* to p e occurs outside the tube, in the 
surrounding medium. In other words, the pressure drop along the tube 
cannot be greater than from_p to/)*, whatever the external pressure. For air 
(p* = 0-53p ), the maximum pressure drop is 047/> . The velocity at the 
end of the tube and the discharge also remain constant for p e < p*. Thus the 
gas cannot acquire a supersonic velocity in flowing through a nozzle of this 
kind. 




Fig. 54 



'£&//////< 




Fig. 55 



If we consider only the flow in the immediate neighbourhood of the end of 
the tube, the motion of the gas after leaving the tube is essentially flow round 
an angle, the vertex of which is the edge of the tube mouth; we shall discuss 
this flow in detail in §104. 

The impossibility of achieving supersonic velocities by flow through a 
continually narrowing nozzle is due to the fact that a velocity equal to the 
local velocity of sound can be reached only at the very end of such a tube. It 
is clear that a supersonic velocity can be attained by means of a nozzle which 
first narrows and then widens again (Fig. 55). This is called a de Laval 
nozzle. 



§90 



Flow of gas through a nozzle 



349 



The maximum flux density j%, if reached, can again occur only at the 
narrowest cross-section, so that the discharge cannot exceed S mi3 J^. In the 
narrowing part of the nozzle, the flux density increases (and the pressure 
falls); the curve in Fig. 56 shows j as a functionf of^>, and the variation just 
described corresponds to the interval from c to b. If the maximum flux 
density is reached at the cross-section S mln (the point b in Fig. 56), the pres- 
sure continues to diminish in the widening part of the nozzle, while^' begins to 
decrease also, corresponding to the segment ba of the curve. At the outer 
end of the tube; takes a definite value, / lmax = /* S ml JS h and the pressure 
has the corresponding value, denoted in Fig. 56 by^i', at some point d on the 
curve. If, however, only some point e is reached at the cross-section S mln , 
the pressure increases in the widening part of the nozzle, corresponding to a 
return down the curve from e towards c. At first sight it might appear that we 
might pass discontinuously from cb to ab, without going through the point b, 
by the formation of a shock wave. This, however, is impossible, since the 
gas "entering" the shock wave cannot have a subsonic velocity. 




Bearing in mind these results, let us now investigate the manner of variation 
in the outflow when the external pressure p e is gradually increased. For small 
pressures, from zero to pi, the pressure p* and velocity v* = c* are reached 
at the cross-section ,S min . In the widening part of the nozzle the velocity 
continues to increase, so that there results a supersonic flow of the gas, and 
the pressure accordingly continues decreasing, reaching the value pi at 
the outer end of the tube, whatever the pressure p e . The pressure falls from 
pi top e outside the nozzle, in the rarefaction wave which leaves the edge of 
the tube mouth (see §104). 

When p e exceeds pi, an oblique shock wave leaves the edge of the tube 
mouth, compressing the gas from pi to p e (§104). We shall see, however, 
that a steady shock wave can leave a solid surface only if its intensity is not too 



f According to formulae (80.15-80.17), the dependence is 



-(s) 



2y 



y-1 



popo 



[-( 



.(y-l)/y 



po) II ' 



11*. 



350 One-dimensional Gas Flow §91 

great (§103). Hence, when the external pressure increases further, the shock 
wave soon begins to move into the nozzle, with separation occurring in front 
of it on the inner surface of the tube. For some value of p e the shock wave 
reaches the narrowest cross-section and then disappears; the flow becomes 
everywhere subsonic, with separation on the walls of the widening part of 
the nozzle. All these complex phenomena are, of course, three-dimensional. 

PROBLEM 

A small amount of heat is supplied over a short segment of a tube to a perfect gas in steady 
flow in the tube. Determine the change in the gas velocity when it passes through this seg- 
ment. 

Solution. Let Sq be the amount of heat supplied per unit time, S being the cross-sectional 
area of the tube at the segment concerned. The mass flux density j = pv and the momentum 
flux density p+jv are the same on both sides of the heated segment; hence A/> = — jAy, 
where A denotes the change in a quantity in passing through the segment. The difference in 
the energy flux density (w+fr^j is q. Writing to = y/>/(y-l)p = ypvl(y-\)j, we obtain 
(supposing Av and Ap small) vjAv + y(pAv+vAp)f(y-l) = q. Eliminating Ap, we find 
A„ _ (y—l)q/p(c 2 —v 2 ). We see that, in subsonic flow, the supply of heat accelerates the 
flow (Av > 0), while in supersonic flow it retards it. 

Writing the gas temperature asT= ftpJRp = p-pv/Rj (R being the gas constant), we find 

For supersonic flow, this expression is always positive, and the gas temperature is increased ; 
for subsonic flow, however, AT may be either positive or negative. 

§91. Flow of a viscous gas in a pipe 

Let us consider the flow of a gas in a pipe (of constant cross-section) so 
long that the friction of the gas against the walls, i.e. the viscosity of the gas, 
cannot be neglected. We shall suppose the walls to be thermally insulated, so 
that there is no heat exchange between the gas and the surrounding medium. 

For gas velocities of the order of or exceeding the velocity of sound (the 
only case we shall discuss here), the gas flow in the pipe is, of course, turbulent 
if the radius of the pipe is not small. The turbulence of the flow is important, 
as regards our problem, only in one respect: we have seen in §43 that, in 
turbulent flow, the (mean) velocity is practically the same almost everywhere 
in the cross-section of the pipe, and falls rapidly to zero very close to the 
walls. We shall therefore suppose that the gas velocity v is a constant over 
the cross-section, and define it so that the product Spv (S being the cross- 
sectional area) is equal to the total discharge through the cross-section. 

Since the total discharge Spv is constant along the pipe, and S is assumed 
constant, the mass flux density must also be constant: 

j = pv — constant. (91.1) 

Next, since the pipe is thermally insulated, the total energy flux carried by 
the gas through any cross-section must also be constant. This flux is 
Spv(w + ^v 2 '), and by (91.1) we have 

w +%v 2 = w+^V 2 = constant. (91.2) 



§91 Flow of viscous gas in a pipe 351 

The entropy s of the gas does not, of course, remain constant, but increases 
as the gas moves along the pipe, because of the internal friction. If x is 
the co-ordinate along the pipe, with x increasing downstream, we can write 

dsjdx > 0. (91.3) 

We now differentiate (91.2) with respect to x. Since dzv = Tds+ Vdp, we 

have 

ds dp dV 

T T +V T + J 2V -J- = - 
dx dx dx 



Next, substituting 



dV 

dx 



we obtain 



/ dV\ dp / 8V\ ds 
= \~di) s dx + \17) p dx' (91 * 4) 

h'^Js-^h^JS (9L5) 

By a well-known formula of thermodynamics, (dV/ds) p = (TJc p )(dVI8T) p . 
The coefficient of thermal expansion is positive for gases. We therefore 
conclude, using (91.3), that the left-hand side of (91.5) is positive. The sign 
of the derivative dpjdx is therefore that of -[l+j 2 (dVldp) s ] = (vjcf-\. 
We see that 

dp/d*§0 for v$c. (91.6) 

Thus, in subsonic flow, the pressure decreases downstream, as for an in- 
compressible fluid. For supersonic flow, however, it increases. 

We can similarly determine the sign of the derivative dvjdx. Since 
j = vjV = constant, the sign of da/cbc is the same as that of dVJdx. The 
latter can be expressed in terms of the positive derivative dsjdx by means of 
(91.4) and (91.5). The result is that 

dvjdx^O for v$c, (91.7) 

i.e. the velocity increases downstream for subsonic flow and decreases for 
supersonic flow. 

Any two thermodynamic quantities for a gas flowing in a pipe are functions 
of one another, independent of (inter alia) the resistance law for the pipe. 
These functions depend on the constant j as a parameter, and are given by 
the equation w + %j 2 V 2 = constant, which is obtained by eliminating the velo- 
city from the equations of conservation of mass and energy for the gas. 

Let us ascertain the nature of the curves giving, for example, the entropy 
as a function of pressure. Rewriting (91.5) in the form 

ds (vjc) 2 — 1 



dp T+jW(dVfds) p ' 

we see that, at the point where v = c, the entropy has an extremum. It is easy 



352 



One-dimensional Gas Flow 



§91 



to see that s has a maximum. For the second derivative of s with respect to 
p at this point is 



rd% 
Ldp2 



J V=C 



]W{?W\dp*) s 



<0; 



T+pV(dVl8s) p 

we assume, as usual, that the derivative {d 2 Vjdp 2 ) s is positive. 

The curves giving $ as a function of p (called Fanno lines) are therefore as 
shown in Fig. 57. The region of subsonic velocities lies to the right of the 
maximum, and that of subsonic velocities to the left. When the parameter j 
increases, we go to lower curves. For, differentiating equation (91.2) with 
respect to j for constant p, we have 

ds _ jV* 

d/ = " T+jW(dV/ds) p 



< 0. 




Fig. 57 



We can draw an interesting conclusion from the above results. Let the gas 
velocity at the entrance to the pipe be less than that of sound. The entropy 
increases downstream, and the pressure decreases; this corresponds to a 
movement along the right-hand branch of the curve s = s(p), from B to- 
wards O (Fig. 57). This can, however, continue only until the entropy reaches 
its maximum value. A further movement along the curve beyond O (i.e. into 
the region of supersonic velocities) is not possible, since the entropy of the 
gas would have to decrease as it moved along the pipe. The transition be- 
tween the branches BO and OA cannot even be effected by a shock wave, 
since the gas entering a shock wave cannot move with subsonic velocity. 

Thus we conclude that, if the gas velocity at the entrance to the pipe is less 
than that of sound, the flow remains subsonic everywhere in the pipe. The 
gas velocity becomes equal to the local velocity of sound only at the other end 
of the pipe, if at all (it does so if the pressure of the external medium into 
which the gas issues is sufficiently low). 



§92 One-dimensional similarity flow 353 

In order that the gas should have supersonic velocities in the pipe, its 
velocity at the entrance must be supersonic. By the general properties of 
supersonic flow (the impossibility of propagating disturbances upstream), 
the flow will then be entirely independent of the conditions at the outlet of 
the pipe. In particular, the entropy will increase along the pipe in a quite 
definite manner, and its maximum value will be attained at a definite distance 
x = h from the entrance. If the total length / of the pipe is less than 4, the 
flow is supersonic throughout the pipe (corresponding to movement on the 
branch AO from A towards O). If, on the other hand, / > l k , the flow cannot 
be supersonic throughout the pipe, nor can there be a smooth transition to 
subsonic flow, since we can move along the branch OB only in the direction 
shown by the arrow. In this case, therefore, a shock wave must necessarily 
be formed, which discontinuously changes the flow from supersonic to sub- 
sonic. The pressure is thereby increased, and we pass from the branch AO 
to BO without going through the point O. The flow is entirely subsonic 
beyond the discontinuity. 

§92. One-dimensional similarity flow 

An important class of one-dimensional non-steady gas flows is formed by 
flows occurring in conditions where there are characteristic velocities but 
not characteristic lengths. The simplest example of such a flow is given by 
gas flow in a semi-infinite cylindrical pipe terminated by a piston, when the 
piston begins to move with constant velocity. 

Such a flow is defined by the velocity parameter and by parameters which 
give, say, the gas pressure and density at the initial instant. We can, however, 
form no combination of these parameters which has the dimensions of length 
or time. It therefore follows that the distributions of all quantities can depend 
on the co-ordinate x and the time t only through the ratio xft, which has the 
dimensions of velocity. In other words, these distributions at various in- 
stants will be similar, differing only in the scale along the x-axis, which in- 
creases proportionally to the time. We can say that, if lengths are measured 
in a unit which increases proportionally to t, then the flow pattern does not 
change. When the flow pattern is unchanged with time if the scale of length 
varies appropriately, we speak of a similarity flow. 

The equation of conservation of entropy for a flow which depends on only 
one co-ordinate, *, is dsjdt + Vx ds/dx = 0. Assuming that all quantities 
depend only on £ = xjt, and noticing that in this case d/dx = (l/f)d/df, 
djdt = -(£/*)d/d£, we obtain (v x - £) s' = (the prime denoting differen- 
tiation with respect to £). Hence s' = 0, i.e. s = constantf ; thus similarity 
flow in one dimension must be isentropic. Likewise, from the y and z com- 
ponents of Euler's equation: dv y ldt + v x dv y jdx = 0, dv z /dt + v x dv z /dx = 0, 
we find that v y and v z are constants, which we can take as zero without loss 
of generality. 

t The assumption that v x — £ = would contradict the other equations of motion; from (92.3) 
we should have v x = constant, contrary to hypothesis. 



354 One-dimensional Gas Flow §92 

Next, the equation of continuity and the ^-component of Euler's equation 
are 

dp dv dp 

■£ + T + V = ' (92<1) 

dt ox ox 

dv dv 1 dp 

+ V= __Z ; ( 92.2) 

dt ox p ox 

here and henceforward we write v x as v simply. In terms of the variable $ , 
these equations become 

(v-€)p'+pv' = 0, (92.3) 

( V -Qv> = -p'j P = -cy/p. (92.4) 

In the second equation we have putp' = {dp\dp) s p = c 2 p', since the entropy 
is constant. 

These equations have, first of all, the trivial solution v = constant, 
p = constant, i.e. a uniform flow of constant velocity. To find a non-trivial 
solution, we eliminate p and v' from the equations, obtaining (v— £) 2 = c 2 y 
whence | = v± c. We shall take the plus sign: 

x\t = v + c; (92.5) 

this choice of sign means that we take the positive #-axis in a definite direction, 
selected in a manner shown later. Finally, putting v— £ = — c in (92.3), 
we obtain cp = pv', or pdv = cdp. The velocity of sound is a function of 
the thermodynamic state of the gas; taking as the fundamental thermodyna- 
mic quantities the entropy s and the density />, we can represent the velocity 
of sound as a function c(p) of the density, for any given value of the constant 
entropy. With c understood as such a function, we can write 

v = j cdp/p = j dpjcp. (92.6) 

This formula can also be written 

v = j ^/(-dpdV), (92.7) 

in which the choice of dependent variable remains open. 

Formulae (92.5) and (92.6) give the required solution of the equations of 
motion. If the function c(p) is known, then the velocity v can be calculated 
as a function of density from (92.6). Equation (92.5) then determines the 
density as an implicit function of xjt, and so the dependence of all the other 
quantities on xjt is determined also. 

We can derive some general properties of the solution thus obtained. 
Differentiating equation (92.5) with respect to x, we have 

dp d(v + c) 
t— \ = 1. (92.8) 

dx dp 



§92 One-dimensional similarity flow 355 

For the derivative of v + c we have, by (92.6), 

d(v + c) c dc 1 d(pc) 
dp p dp p dp 

But 

P c = pvWp) = W(-W; 

differentiating, we have 

dOoc)/dp = &l(pc)ldp = lp*(*(d*VldpP) 8 . (92.9) 

Thus 

d(© + c)/dp = y 2 c%8W/dp^) s > 0. (92.10) 

It therefore follows from (92.8) that dp/dx > for t > O.f Since dp/dx 
= c 2 dp/dx, we conclude that dp/&* > also. Finally, we have dvjdx 
= (c/p)dp/dx, so that dvjdx > 0. The inequalities 

8p/8x > 0, dp/dx > 0, dv/dx > (92.11) 

therefore hold. 

The meaning of these inequalities becomes clearer if we follow the variation 
of quantities, not along the x-axis for given t, but with time for a given gas 
element as it moves about. This variation is given by the total time deriva- 
tive ; for the density, for example, we have, using the equation of continuity, 
dp/dt = dp/dt+v dp/dx = —p dv/dx. By the third inequality (92.11), 
this quantity is negative, and therefore so is dp/dt: 

dp/dt < 0, dp/dt < 0. (92.12) 

Similarly (using Euler's equation (92.2)) we can see that dv/dt < 0; this, 
however, does not mean that the magnitude of the velocity diminishes with 
time, since v may be negative. 

The inequalities (92.12) show that the density and pressure of any gas 
element decrease as it moves. In other words, the gas is continually rarefied 
as it moves. Such a flow may therefore be called a non-steady rarefaction 
wave. 

A rarefaction wave can be propagated only a finite distance along the #-axis ; 
this is seen from the fact that formula (92.5) would give an infinite velocity 
for x -> + oo, which is impossible. 

Let us apply formula (92.5) to a plane bounding the region of space occupied 
by the rarefaction wave. Here x/t is the velocity of this boundary relative to 
the fixed co-ordinate system chosen. Its velocity relative to the gas itself is 
(x/t) — v and is, by (92.5), equal to the local velocity of sound. This means 
that the boundaries of a rarefaction wave are weak discontinuities. The 



t There is no meaning for times t < in the similarity flow here considered. Such a flow can 
occur only because of some singularity in the initial conditions (t = 0) of the flow at the point x = 0, 
and therefore takes place only for t > (in our example, the piston velocity changes discontinuously 
at t - 0. See also §93). 



356 One-dimensional Gas Flow §92 

similarity flow in different cases is therefore made up of rarefaction waves and 
regions of constant flow, separated by surfaces of weak discontinuity. - ]- 

The choice of sign in (92.5) is now seen to correspond to the fact that these 
weak discontinuities are assumed to move in the positive ^-direction relative 
to the gas. The inequalities (92.11) arise from this choice, but the inequalities 
(92.12), of course, do not depend on the direction of the #-axis. 



r 
-J 
m ! 



I 
I 

j 



IT 



Fig. 58 



We are usually concerned, in actual problems, with a rarefaction wave 
bounded on one side by a region where the gas is at rest. Let this region (I 
in Fig. 58) be to the right of the rarefaction wave. Region II is the rarefaction 
wave, and region III contains gas moving with constant velocity. The arrows 
in the figure show the direction of motion of the gas, and of the weak dis- 
continuities bounding the rarefaction wave; the discontinuity a always 
moves into the gas at rest, but the discontinuity b may move in either direction, 
depending on the velocity reached in the rarefaction wave (see Problem 2). 
We may give explicitly the relations between the various quantities in such a 
rarefaction wave, assuming that we have a perfect gas. For an adiabatic 
process pT ll(1 ~ y) = constant. Since the velocity of sound is proportional to 
\/T, we can write this relation as 

P = po{chf'^\ (92.13) 

Substituting this expression in the integral (92.6), we obtain 

2 f 2 

v = dc = -(c-co); 

y—l J y— 1 

the constant of integration is chosen so that c = Co for v = (we use the 
suffix to refer to the point where the gas is at rest). We shall express all 
quantities in terms of v, bearing in mind that, with the above situation of the 
various regions, the gas velocity is in the negative ^-direction, i.e. v < 0. 
Thus 

c = c Q -\{y-\)\v\, (92.14) 

which determines the local velocity of sound in terms of the gas velocity. 
Substituting in (92.13), we find the density to be 

P = Po[l-¥y-l)\v\lco]Wy-», (92.15) 



f There may also, of course, be regions of constant flow separated by shock waves. 



§92 



One-dimensional similarity flow 357 



and similarly the pressure is 

P = Mi -\{y~ i)M M)] 2 ^- 1 *. (92.16) 

Finally, substituting (92.14) in formula (92.5), we obtain 



2 / x 

M = 7 Ko - - 

1 ' y+l\ t 



(92.17) 



which gives a as a function of # and t. 

The quantity c cannot be negative, by definition. We can therefore draw 
from (92.14) the important conclusion that the velocity must satisfy the 
inequality 

M <2* /(y-l); ( 92 - 18 ) 

when the velocity reaches this limiting value, the gas density (and also/) and 
c) becomes zero. Thus a gas originally at rest and expanding non-steadily 
in a rarefaction wave can be accelerated only to velocities not exceeding 

2*o/(y-l). . . r L . • • , 

We have already mentioned, at the beginning of this section, a simple 

example of similarity flow, namely that which occurs in a cylindrical pipe in 
which a piston begins to move with constant velocity. If the piston moves out 
of the pipe, it creates a rarefaction, and a rarefaction wave of the kind des- 
cribed above is formed. If, however, the piston moves inwards, it compresses 
the gas in front of it, and the transition to the original lower pressure can occur 
only in a shock wave, which is in fact formed in front of a piston moving for- 
ward in a pipe (see the following Problems)-! 

PROBLEMS 

Problem 1. A perfect gas occupies a semi-infinite cylindrical pipe terminated by a 
piston. At an initial instant the piston begins to move into the pipe with constant velocity U. 
Determine the resulting flow. 

Solution A shock wave is formed in front of the piston, and moves along the pipe. 
At the initial instant this shock and the piston are coincident, but at subsequent instants the 
shock is ahead of the piston, and a region of gas lies between them (region 2). In front or 
the shock wave (region 1), the gas pressure is equal to its initial value p u and its velocity 
relative to the pipe is zero. In region 2, the gas moves with constant velocity, equal to the 
velocity U of the piston (Fig. 59). The difference in velocity between regions 1 and 2 is 
therefore also U, and, by formulae (82.7) and (85.1), we can write 

u= V[(p2-piWi-v 2 )] 

= (p2-piW{2Vil[(y-l)Pi + (y+^)p2]y 



t We may mention also an analogous similarity flow in three dimensions: the centrally symmetrical 
sas flow caused by a uniformly expanding sphere. A spherical shock wave, expanding with constant 
velocity is formed in front of the sphere. Unlike what happens in the one-dimensional case, the 
velocity of the gas between the sphere and the shock is not constant; the equation which determines 
it as a function of the ratio r\t (and therefore the rate of propagation of the shock wave) cannot be 

m Twf ptoWemha^been discussed by L. I. SedoV (1945; see his book Similarity and Dimensional 
Methods in Mechanics, Cleaver-Hume Press, London 1959) and by G. I. Taylor, Proceedings of the 
Royal Society, A186, 273, 1946. 



358 



One-dimensional Gas Flow 



§92 



Hence we find the gas pressure p z between the piston and the shock wave to be given by 

( y+ l)2f/2 

— X T " 1 / 1 -t- 

Pi 



4c,2 



Cl 



\6a* 



Knowing p 2 , we can calculate, from formulae (85.4), the velocity of the shock wave relative 
to the gas on each side of it. Since gas 1 is at rest, the velocity of the shock relative to it is 
equal to the rate of propagation of the shock in the pipe. If the x co-ordinate (along the pipe) 
is measured from the initial position of the piston (the gas being on the side x > 0), we find 
the position of the shock wave at time t to be 

while the position of the piston is x = Ut. 




t 



U 
Fig. 59 



(a) ^ 



2 3 



-U 



(fo-^0) f 




Problem 2. The same as Problem 1, but for the case where the piston moves out of the 
pipe with velocity U. 

Solution. The piston adjoins a region of gas (region 1 in Fig. 60a) which moves in the 
negative ^-direction with constant velocity — U, equal to the velocity of the piston. Then 
follows a rarefaction wave (2), in which the gas moves in the negative ^-direction, its velocity 
varying linearly from — U to zero according to (92.17). The pressure varies according to 
(92.16) from p x = £ [l_£( y _l)£// Co ]2y/ ( y-i) in gas j to Po in the gas 3> which fg &t res ^ 

The boundary of regions 1 and 2 is given by the condition v — —U; according to (92.17), 
we have x = [c —}(y+l)U]t = (c—U)t, where c is the velocity of sound in gas 1. At the 
boundary of regions 2 and 3, v = 0, whence * = c t. Both boundaries are weak discon- 
tinuities ; the second is always propagated to the right (i.e. away from the piston), but the first 
may be propagated either to the right (as shown in Fig. 60a) or to the left (if the piston 
velocity U > 2c /(y+l)). 



§92 



One-dimensional similarity flow 



359 



The flow pattern just described can occur only if U < 2c /(y — 1). If U > 2c„/(y — 1), a 
vacuum is formed in front of the piston (the gas cannot follow the piston), which extends 
from the piston to the point x = -2c «/(y-l) (region 1 in Fig. 60b). At this point, 
v = -2c /(y-l); then follow region 2, in which the velocity decreases to zero at the point 
x = c t, and region 3, where the gas is at rest. 

Problem 3. A gas occupies a semi-infinite cylindrical pipe (x > 0) terminated by a valve. 
At time t = 0, the valve is opened, and the gas flows into the external medium, the pressure 
pe in which is less than the initial pressure p in the pipe. Determine the resulting flow. 




Fig. 61 



Solution. Let — v e be the gas velocity which corresponds to the external pressure p e 
according to formula (92.16); for * = and t > 0, we must have v = -v e . If 
v e < 2c /(y+l), the velocity distribution shown in Fig. 61a results. For v e = 2c /(y+l) 
(corresponding to a rate of outflow equal to the local velocity of sound at the end of the pipe : 
this is easily seen by putting v = c in formula (92.14)), the region of constant velocity vanishes 
and the pattern shown in Fig. 61b is obtained. The quantity 2c /(y + l) is the greatest 
possible rate of outflow from the pipe in the conditions stated. If the external pressure pe 
is such that __„ . . /1N 

p e </>o[2/(y + l)] 2 >' /( ^ 1) , (1) 

the corresponding velocity exceeds 2c /(y+l). In reality, the pressure at the pipe outlet 
would still be equal to the limiting value (the right-hand side of (1)), and the rate of outflow 
would be 2c„/(y+l); the remaining pressure drop (to p e ) occurs in the external medium. 

Problem 4. An infinite pipe is divided by a piston, on one side of which (x < 0) there is, 
at the initial instant, gas at pressure p , and on the other side a vacuum. Determine the motion 
of the piston as the gas expands. 

Solution. A rarefaction wave is formed in the gas ; one of its boundaries moves to the right 
with the piston, and the other moves to the left. The equation of motion of the piston is 

mdU/dt = po[\-\{y-\)Ujc Q fy^^\ 

where U is the velocity of the piston and m its mass per unit area. Integrating, we obtain 



™-SH'^] 



(y+l)po-\^- my+1) \ 



Problem 5. Determine the flow in an isothermal similarity rarefaction wave. 

Solution. The isothermal velocity of sound is ct = V(dp/dp)T = V(RTffi), and for 
constant temperature ct = constant = cto- According to (92.5) and (92.6), we therefore 

v = c To log(p/po) = c T ^og{pjpo) = (xji)-c Tt . 



360 One-dimensional Gas Flow §93 

§93. Discontinuities in the initial conditions 

One of the most important reasons for the occurrence of surfaces of dis- 
continuity in a gas is the possibility of discontinuities in the initial conditions. 
These conditions (i.e. the initial distributions of velocity, pressure, etc.) 
may in general be prescribed arbitrarily. In particular, they need not be 
everywhere continuous, but may be discontinuous on various surfaces. For 
example, if two masses of gas at different pressures are brought together at 
some instant, their surface of contact will be a surface of discontinuity of the 
initial pressure distribution. 

It is of importance that the discontinuities of the various quantities in the 
initial conditions (or, as we shall say, in the initial discontinuities) can have 
any values whatever; no relation between them need exist. We know, how- 
ever, that certain conditions must hold on stable surfaces of discontinuity in a 
gas; for instance, the discontinuities of density and pressure in a shock wave 
are related by the shock adiabatic. It is therefore clear that, if these conditions 
are not satisfied in the initial discontinuity, it cannot continue to be a dis- 
continuity at subsequent instants. Instead, the initial discontinuity in general 
splits into several discontinuities, each of which is one of the possible types 
(shock wave, tangential discontinuity, weak discontinuity) ; in the course of 
time, these discontinuities move apart. A general discussion of the behaviour 
of an arbitrary discontinuity has been given by N. E. Kochin (1926). 

During a short interval of time after the initial instant t = 0, the discon- 
tinuities formed from the initial discontinuity do not move apart to great 
distances, and the flow under consideration therefore takes place in a relatively 
small volume adjoining the surface of initial discontinuity. As usual, it 
suffices to consider separate portions of this surface, each of which may be 
regarded as plane. We need therefore consider only a plane surface of 
discontinuity, which we take as the yz-plzne. It is evident from symmetry 
that the discontinuities formed from the initial discontinuity will also be plane, 
and perpendicular to the *-axis. The flow pattern will depend on the co- 
ordinate x only (and on the time), so that the problem is one-dimensional. 
There being no characteristic parameters of length and time, we have a 
similarity problem, and the results obtained in §92 can be used. 

The discontinuities formed from the initial discontinuity must evidently 
move away from their point of formation, i.e. away from the position of the 
initial discontinuity. It is easy to see that either one shock wave, or one pair 
of weak discontinuities bounding a rarefaction wave, can move in each direc- 
tion (the positive and negative ^-direction). For, if there were, say, two shock 
waves formed at the same point at time t = and both propagated in the 
positive x-direction, the leading one would have to move more rapidly than 
the other. According to the general properties of shock waves, however, the 
leading shock wave must move, relative to the gas behind it, with a velocity 
less than the velocity of sound c in that gas, and the following shock must 
move, relative to the same gas, with a velocity exceeding c (c being a constant 
in the region between the shock waves), i.e. it must overtake the other. For 



§93 Discontinuities in the initial conditions 361 

the same reason, a shock wave and a rarefaction wave cannot move in the same 
direction; to see this, it is sufficient to notice that weak discontinuities move 
with the velocity of sound relative to the gas on each side of them. Finally, 
two rarefaction waves formed at the same time cannot become separated, 
since the velocities of their backward fronts are the same. 

As well as shock waves and rarefaction waves, a tangential discontinuity 
must in general be formed from an initial discontinuity. Such a discontinuity 
must occur if the transverse velocity components v y , v z are discontinuous in 
the initial discontinuity. Since these velocity components do not change in a 
shock or rarefaction wave, their discontinuities always occur at a tangential 
discontinuity, which remains at the position of the initial discontinuity; on 
each side of this discontinuity, v y and v z are constant (in reality, of course, 
the instability of a tangential velocity discontinuity causes its gradual smooth- 
ing into a turbulent region). 

A tangential discontinuity must occur, however, even if v y and v z are 
continuous at the initial discontinuity (without loss of generality, we can, and 
shall, assume that they are zero). This is shown as follows. The discon- 
tinuities formed from the initial discontinuity must make it possible to go from 
a given state 1 of the gas on one side of the initial discontinuity to a given state 
2 on the other side. The state of the gas is determined by three independent 
quantities, e.g. p, p and v x = v. It is therefore necessary to have three arbi- 
trary parameters in order to go from state 1 to an arbitrary state 2 by some 
choice of the discontinuities. We know, however, that a shock wave, per- 
pendicular to the stream, propagated in a gas whose thermodynamic state is 
given, is completely determined by one parameter (§82). The same is true of 
a rarefaction wave; as we see from formulae (92.14)-(92.16), when the state 
of the gas entering a rarefaction wave is given, the state of the gas leaving it is 
completely determined by one parameter. We have seen, moreover, that at 
most one wave (rarefaction or shock) can move in each direction. We therefore 
have at our disposal only two parameters, which are not sufficient. 

The tangential discontinuity formed at the position of the initial discon- 
tinuity furnishes the third parameter required. The pressure is continuous 
there, but the density (and therefore the temperature and entropy) is not. 
The tangential discontinuity is stationary with respect to the gas on both sides 
of it and the arguments about the "overtaking" of two waves propagated in the 
same direction therefore do not apply to it. 

The gases on the two sides of the tangential discontinuity do not mix, 
since there is no motion of gas through a tangential discontinuity; in all the 
examples given below, these gases may be different substances. 

Fig. 62 shows schematically all possible types of break-up of an initial 
discontinuity. The continuous line shows the variation of the pressure along 
the #-axis; the variation of the density would be given by a similar line, the 
only difference being that there would be a further jump at the tangential 
discontinuity. The vertical lines show the discontinuities formed, and the 
arrows show their direction of propagation and that of the gas flow. The 



362 



One-dimensional Gas Flow 



§93 



co-ordinate system is always that in which the tangential discontinuity is at 
rest, together with the gas in the regions 3 and 3' which adjoin it. The pres- 
sures, densities and velocities of the gases in the extreme left-hand (1) and 
right-hand (2) regions are the values of these quantities at time t = on each 
side of the initial discontinuity. 



3 3 









2 


1 






— 


-* 


*~S+TS^ , % 







(a) 





3- 


3' 


4/| 


1 •"• 






**" ! 


— 





I-*-S^_TR_ 



(b) 



1 




3' 


r ! 




- 


1 • ' 

— i — i i_ 





1 + R^.TR 



(c) 




Z-W? 



r "*■ (d) 
Shock wave 



Tangential discontinuity 

■- Weak discontinuity 

Fig. 62 



In the first case, which we write I -> S^ TS_* (Fig. 62a), the initial dis- 
continuity / gives two shock waves S, propagated in opposite directions, 
and a tangential discontinuity T between them. This case occurs when two 
masses of gas collide with a large relative velocity. 

In the case / -> fi_ TR_> (Fig. 62b), a shock wave is propagated on one 
side of the tangential discontinuity, and a rarefaction wave R on the other 
side. This case occurs, for instance, if two masses of gas at relative rest 
(v2 — V! = 0) and at different pressures are brought into contact at the initial 



§93 Discontinuities in the initial conditions 363 

instant. For, of all the cases shown in Fig. 62, the second is the only one in 
which gases 1 and 2 are moving in the same direction, and so the equation 
vi — v 2 is possible. . 

In the third case (/ -> i?_ TR^, Fig. 62c), a rarefaction wave is propagated 
on each side of the tangential discontinuity. If gases 1 and 2 separate with a 
sufficiently great relative velocity v 2 -v x , the pressure may decrease to zero 
in the rarefaction waves. We then have the pattern shown in Fig. 62d; a 
vacuum 3 is formed between regions 4 and 4'. 

We can derive the analytical conditions which determine the manner in 
which the initial discontinuity breaks up, as a function of its parameters. We 
shall suppose in every case that£ 2 > pi, and take the positive ^-direction from 
region 1 to region 2 (as in Fig. 62). 

Since the gases on the two sides of the initial discontinuity may be ot 
different substances, we shall distinguish them as gases 1 and 2. 

(1) / -* S<_TS^. If pz = pz', Vz and Vz' are the pressures and specific 
volumes in the resulting regions 3 and 3', then we have pz>pz> pi, and the 
volumes Vz and V Z ' are the abscissae of the points with ordinate pz on the 
shock adiabatics through (p lf Vi) and (p 2 , V 2 ) respectively. Since the gases 
in regions 3 and 3' are at rest in the co-ordinate system chosen, we can use 
formula (82.7) to give the velocities v x and v 2 , which are in the positive and 
negative ^-directions respectively: 

vi = Vlipz-piWi- v *)l v 2 = - V[(p8-^2)(^2- Vz')]. 
The least value of pz, for given pi and p 2 , which does not contradict the initial 
assumption (p 3 > p2 > pi) is pi- Since, moreover, the difference ^ - c* is a 
monotonically increasing function of p z , we find the required inequality 

vi-v 2 > Vm-pi)(Vi- V')], (93.1) 

where V denotes the abscissa of the point with ordinate p 2 on the shock 
adiabatic for gas 1 through (/>i, Vi). Calculating V from formula (85.1) (in 
which V 2 is replaced by V), we obtain the condition (93.1) for a perfect gas 
in the form 

V!-V2 > (P2-/>i)V(2M(n-i)/>i + (n + i)KI}- ( 93 - 2 ) 

It should be noted that the limits placed by (93.1) and (93.2) on the possible 
values of the velocity difference v 1 -v 2 clearly do not depend on the co-ordi- 
nate system chosen. 

(2) / -> S^TR^. Here^i <pz =pz'<P* For the gas velocity m region 1 

we again have 

*>i = y/[{pz-piWi-Vz)l 
and the total change in velocity in the rarefaction wave 4 is, by (92.7), 

Pi 
v 2 = J V(-#dF). 

P* 



364 One-dimensional Gas Flow §93 

For given p x and p 2 , p 3 can lie between them. Replacing p z in the difference 
vz-vx by pi and then by p 2 , we obtain the condition 



- j V(-dpdV) < V!-v 2 < V[(pz-pi){Vi-V')]. 



(93.3) 



Here V has the same significance as in the previous case; the upper limit of 
the difference V! - v 2 must be calculated for gas 1, and the lower limit for gas 2. 
For a perfect gas we have 

2c 2 r /p 1 \(y.-i)/2y,i 

< (P2 -pi)V{2Vi/[(n - l)^i + (yi + l)pz]}, (93.4) 

where c 2 = ^/(y 2 p 2 V 2 ) is the velocity of sound in gas 2 in the state (p 2 , V 2 ). 
(3) / -> R^TR^. Here p 2 > p 1 >p 3 = p z > > 0. By the same method we 
find the following condition for this case to occur: 



- j V(-dpdV)- J V(-dpdV) < v x -v 2 < - j ^/(-dpdV). (93.5) 



The first integral in the first member is calculated for gas 1, and the others 
for gas 2. For a perfect gas we find 

2ci 2c 2 2c 2 r / p x \ (y s -i)/2y 2 -i 

7 -<V!-v 2 < h_(£_ I (93. 6 ) 

yi-1 y 2 -\ 72-1 L \p 2 ] y K ' 

where c ± = Vinpi^i), c 2 = V(72p2V 2 ). If 

2c± 2c 2 

vi-v 2 < -, (93.7) 

yi-1 y2-l 

a vacuum is formed between the rarefaction waves (/ -> R+_ VRJ). 

The problem of a discontinuity in the initial conditions includes that 
of various collisions between plane surfaces of discontinuity. At the instant of 
collision, the two planes coincide, and form some initial discontinuity, which 
then leads to one of the patterns described above. The collision of two shock 
waves, for instance, results in two other shock waves, which move away 
from the tangential discontinuity remaining between them : S^S*. -» S+. TS^. 
When one shock wave overtakes another, there are two possibilities: S_*S^ 
-> S±_ TS_> and S_>S^ -> R^ TS^. In either case a shock wave continues in the 
same direction. 

The problem of the reflection and transmission of shock waves by a tan- 
gential discontinuity (boundary of two media) also comes under this heading. 
Here two cases are possible: S^T-+S<.TS_ and S^T->R+_TS_+. The wave 



§93 



Discontinuities in the initial conditions 



365 



transmitted into the second medium is always a shock (see also the following 
Problems)-! 

PROBLEMS 

Problem 1. A plane shock wave is reflected from a rigid plane surface. Determine the gas 
pressure behind the reflected wave (S. V. Izmailov 1935). 



*■ 1 



Fig. 63 



/, 



Solution When a shock wave is incident on a rigid wall, a reflected shock wave is pro- 
pagated away from the wall. We denote by the suffixes 1, 2 and 3 respectively quantities 
pertaining to the undisturbed gas in front of the incident shock, the gas behind this shock 
(which is also the gas in front of the reflected shock) and the gas behind the reflected shock; 
see Fig. 63, where the arrows indicate the direction of motion of the shock waves and of the 
gas itself. The gas in regions 1 and 3, which adjoin the wall, is at rest relative to the wall. 
The relative velocity of the gases on the two sides of the discontinuity is the same in both 
the incident and the reflected shock wave, and equal to the velocity of gas 2. Using formula 
(82.7) for the relative velocity, we therefore have (p 2 — £i)(I / i — V 2 ) = (p 3 — p2)(.V 2 — V 3 ). The 
equation of the shock adiabatic (85.1) for each shock gives 



Yl 



(y+l)/>i+(y-l)/>2 



Yl 

v 2 



(y+l)/>2+(y-l)/* 



(y-l)/>i + (y+l)/>2 V 2 (y-l)p2 + (y+l)^ 3 

We can eliminate the specific volumes from these three equations, and the result is 

(/>3-/>2) 2 [(y+l)£i + (y-l)/>2] = (p2-£i) 2 [(y+l)/>3+(y-l)M 

This is a quadratic equation for p3, which has the trivial root p 3 = pi, cancelling p 3 — pu 
we obtain 



p3_ = (3y-l)j>2-(y-l)j>i 
P2 ~ (y-l)/>2-(y+l)^i' 



which determines p 3 from p x and p t . In the limiting case of a very strong incident shock, 
p 3 = (3y-l)£ 2 /(y-l), while for a weak shock p 3 -p 2 = Pz-pi, corresponding to the sound- 
wave approximation. 

Problem 2. Find the condition for a shock wave to be reflected from a plane boundary 
between two gases. 



| For completeness we should mention that, when a shock wave collides with a weak discontinuity 
(a problem which is not of the similarity type considered here), the shock wave continues to be 
propagated in the same direction, but behind it there remain a weak discontinuity of the original 
kind and a weak tangential discontinuity (see the end of §89). 



366 One-Dimensional Gas Flow §94 

Solution. Let p^ = p r , V u V 2 ', be the pressures and specific volumes of the two media 
before the incidence of the shock wave (propagated in gas 2), at their surface of separation, 
and p 2 ,V 2 the values behind the shock wave. The condition for the reflected wave to be a 
shock wave is given by the inequality (93.2), in which we must now put 



Vl -v 2 = V[(p2-p2')(V 2 '-V 2 )]. 
srms of the ratio of pressures p 2 /pi i 



Expressing all quantities in terms of the ratio of pressures p % \p x and the initial specific volumes 
V u V r , we obtain 



(yi + l)p2lpi + (n- 1) (Y2+l)p2lpi+(n- 1) 

§94. One-dimensional travelling waves 

In discussing sound waves in §63, we assumed the amplitude of oscillations 
in the wave to be small. The result was that the equations of motion were 
linear and were easily solved. A particular solution of these equations is any 
function of x±ct (a plane wave), corresponding to a travelling wave whose 
profile moves with velocity c, its shape remaining unchanged; by the profile 
of a wave we mean the distribution of density, velocity, etc., along the direc- 
tion of propagation. Since the velocity v, the density p and the pressure p 
(and the other quantities) in such a wave are functions of the same quantity 
x±ct y they can be expressed as functions of one another, in which the co- 
ordinates and time do not explicitly appear (p = p(p), v = v(p), and so on). 

When the wave amplitude is not necessarily small, these simple relations 
do not hold. It is found, however, that a general solution of the exact equa- 
tions of motion can be obtained, in the form of a travelling plane wave which is 
a generalisation of the solution f(x ± ct) of the approximate equations valid 
for small amplitudes. To derive this solution, we shall begin from the require- 
ment that, for a wave of any amplitude, the velocity can be expressed as a 
function of the density. 

In the absence of shock waves the flow is adiabatic. If the gas is homo- 
geneous at some initial instant (so that, in particular, s = constant), then 
s = constant at all times, and we shall assume this in what follows. 

In a plane sound wave propagated in the ^-direction, all quantities depend 
on x and t only, and for the velocity we have v x = v, v y = v z = 0. The 
equation of continuity is 8p]8t+8(pv)[8x = 0, and Euler's equation is 

8v 8v 1 8p 

— + v— + - — = 0. 
8t 8x p 8x 

Using the fact that v is a function of p only, we can write these equations 



as 



8p d(pv) 8p 



8v I 1 d*\ 8v 

— + \v +-f- — = 0. (94.2) 

8t \ pdv/8x K ' 



§94 One-dimensional travelling waves 367 

Since 



dpjdt (dx\ 

dp/dx ~ \dt)\ 



we have from (94.1) 

( 

and similarly from (94.2) 



8x \ d(ov) dv 

dt/ p dp dp 

( d _l) =v +\% (94.3) 

\8t/ v p dv 

Since the value of p uniquely determines that of v, the derivatives for con- 
stant p and constant v are the same, i.e. (dx\dt) p = (dxjdt) v , so that p dvldp 
= (1/p) dpldv. Putting dpjdv = (dj>/dp)(d/>/d«>) = ^dp/dv, we obtain dvjdp 
= ± cjp, whence 

r- + f%-±f£ (94-4) 

This gives the general relation between the velocity and the density or pressure 

in the wave.f 

Next, we can combine (94.3) and (94.4) to give (Bxjdt) v = v + {\jp)dpldv 

= v ± c(v), or, integrating, 

x= t[v±c(v)]+f(v), (94.5) 

where f{v) is an arbitrary function of the velocity, and c{v) is given by (94.4). 

Formulae (94.4) and (94.5) give the required general solution (B. Riemann, 
1860). They determine the velocity (and therefore all other quantities) as 
an implicit function of x and t, i.e. the wave profile at every instant. For 
any given value of v, we have * = at + b, i.e. the point where the velocity 
has a given value moves with constant velocity; in this sense, the solution 
obtained is a travelling wave. The two signs in (94.5) correspond to waves 
propagated (relative to the gas) in the positive and negative ^-directions. 

The flow described by the solution (94.4) and (94.5) is often called a 
simple wave, and we shall use this expression below. It should be noticed 
that the similarity flow discussed in §92 is a particular case of a simple wave, 
corresponding to f(v) = in (94.5). 

We can write out explicitly the relations for a simple wave in a perfect 
gas; for definiteness, we assume that there is a point in the wave for which 
v = 0, as usually happens in practice. Since formula (94.4) is the same as 
(92.6), we have by analogy with formulae (92.14)-(92.16) 

c = c ±l(y-l)v, (94.6) 

P = Po(i±Kr-i)^o) 2/<y - 1) , (947) 

/»=Mi±l(r-iW^o) 2 ^- 1) . 

t In a wave of small amplitude we have p = Po +p, and (94.4) gives in the first approximation 
v = Cop'lpo (where c = c(p )), i.e. the usual formula (63.12). 



368 One-dimensional Gas Flow §49 

Substituting (94.6) in (94.5), we obtain 

x=t(± c +i(y+ l)v)+f(v). (94.8) 

It is sometimes convenient to write this solution in the form 

v = F[x-(±c + i(y+l)v)t], (94.9) 

where F is another arbitrary function. 

From formulae (94.6) and (94.7) we again see (as in §92) that the velocity 
in a direction opposite to that of the propagation of the wave (relative to the 
gas itself) is of limited magnitude; for a wave propagated in the positive 
^-direction we have 

-v ^ 2c /(y-l). (94.10) 

A travelling wave described by formulae (94.4) and (94.5) is essentially 
different from the one obtained in the limiting case of small amplitudes. 
The velocity of a point in the wave profile is 

u = v±c\ (94.11) 

it may be conveniently regarded as a superposition of the propagation of a 
disturbance relative to the gas with the velocity of sound and the movement 
of the gas itself with velocity v. The velocity u is now a function of the 
density, and therefore is different for different points in the profile. Thus, 
in the general case of a plane wave of arbitrary amplitude, there is no definite 
constant "wave velocity". Since the velocities of different points in the wave 
profile are different, the profile changes its shape in the course of time. 

Let us consider a wave propagated in the positive ^-direction, for which 
« = v + c. The derivative of v + c with respect to the density has been cal- 
culated in §92; see (92.10). We have seen that du/dp > 0. The velocity of 
propagation of a given point in the wave profile is therefore the greater, the 
greater the density. If we denote by c the velocity of sound for a density 
equal to the equilibrium density p , then in compressions p > p and c> c , 
while in rarefactions p < p and c < c . 

The inequality of the velocity of different points in the wave profile causes 
its shape to change in the course of time: the points of compression move 
forward and those of rarefaction are left behind (Fig. 64b). Finally, the 
profile may become such that the function p{x) (for given t) is no longer 
one-valued; three different values of p correspond to some x (the dashed 
line in Fig. 64c). This is, of course, physically impossible. In reality, 
discontinuities are formed where p is not one- valued, and p is consequently 
one-valued everywhere except at the discontinuities themselves. The wave 
profile then has the form shown by the continuous line in Fig. 64c. The 
surfaces of discontinuity are thus formed at points a wavelength apart. 

When the discontinuities are formed, the wave ceases to be a simple wave. 
The cause of this can be briefly stated thus: when surfaces of discontinuity 
are present, the wave is "reflected" from them, and therefore ceases to be a 



§94 



One-dimensional travelling waves 



369 



wave travelling in one direction. The assumption on which the whole 
derivation is based, namely that there is a one-to-one relation between the 
various quantities, consequently ceases to be valid in general. 

The presence of discontinuities (shock waves) results, as was mentioned in 
§82, in the dissipation of energy. The formation of discontinuities therefore 
leads to a marked damping of the wave. This is evident from Fig. 64. When 
the discontinuity is formed, the highest part of the wave profile is cut off. 
In the course of time, as the profile is bent over, its height becomes less, and 
the profile is "smoothed" to one of smaller amplitude, i.e. the wave is damped. 




Fig. 64 



It is clear from the above that discontinuities must ultimately be formed 
in every simple wave which contains regions where the density decreases in the 
direction of propagation. The only case where discontinuities do not occur 
is a wave in which the density increases monotonically in the direction of 
propagation (such, for example, is the wave formed when a piston moves 
out of an infinite pipe filled with gas; see the Problems at the end of this 
section). 

Although the wave is no longer a simple one when a discontinuity has been 
formed, the time and place of formation of the discontinuity can be deter- 
mined analytically. We have seen that the occurrence of discontinuities is 
mathematically due to the fact that, in a simple wave, the quantities/), p and v 
become many-valued functions of x (for given t) at times greater than a 



370 One-dimensional Gas Flow §94 

certain definite value t , whereas for t < to they are one-valued functions. 
The time *o is the time of formation of the discontinuity. It is evident from 
geometrical considerations that, at the instant t , the curve giving, say, v 
as a function of * becomes vertical at some point x = xo, which is the point 
where the function is subsequently many-valued. Analytically, this means 
that the derivative (8vjdx) t becomes infinite, and (Bxjdv) t becomes zero. It 
is also clear that, at the instant to, the curve v = v(x) must lie on both 
sides of the vertical tangent, since otherwise v(x) would already be many- 
valued. In other words, the point x = x must be, not an extremum of the 
function x(v), but a point of inflexion, and therefore the second derivative 
(d 2 xldv% must also vanish. Thus the place and time of formation of the 
shock wave are determined by the simultaneous equations 

(dx/dv) = 0, (82x[dv2) t = 0. (94.12) 

For a perfect gas these equations are 

* = -2/»/(y+l), /» = 0, (94.13) 

where f(v) is the function appearing in the general solution (94.8). 

These conditions require modification if the simple wave adjoins a gas at 
rest and the shock wave is formed at the boundary. Here also the curve 
v = v{x) must become vertical, i.e. the derivative (dx[8v) t must vanish, at 
the time when the discontinuity occurs. The second derivative, however, 
need not vanish; the second condition here is simply that the velocity is 
zero at the boundary of the gas at rest, so that (dxjdv) t = for v = 0. From 
this condition we can obtain explicit expressions for the time and place of 
formation of the discontinuity. Differentiating (94.5), we obtain 

*= -/'(0)K *= ±c *+/(0), (94.14) 

where ao is the value, for v = 0, of the quantity a defined by formula (95.2). 
For a perfect gas 

*= -2/'(0)/(y+l). (94.15) 

PROBLEMS 

Problem 1 . A perfect gas is in a semi-infinite cylindrical pipe (x > 0) terminated by a piston. 
At time t — the piston begins to move with a uniformly accelerated velocity U = ±at. 
Determine the resulting flow. 

Solution. If the piston moves out of the pipe (U = — at), the result is a simple rare- 
faction wave, whose forward front is propagated to the right, through gas at rest, with 
velocity c ; in the region x > c t the gas is at rest. At the surface of the piston, the gas and 
the piston must have the same velocity, i.e. we must have v = —at for x = —%at 2 (t > 0). 
This condition gives for the function /(*>) in (94.8) 

f(-at) = -cot + %yat*. 

Hence we have 

*-|>o+Ky +1X1* =/(*>) 

= covJa+\yv 2 la t 



§94 



One-dimensional travelling waves 



371 



whence 

-v = [co+Kr + iKI/y- V(h+Ky+ iKI 2 -2«y(<;o*-*)}/y. (l) 

This formula gives the change in velocity over the region between the piston and the forward 
front x = c t of the wave (Fig. 65a) during the time interval t = to t = 2c l(y—l)a. 
The gas velocity is everywhere to the left, like that of the piston, and decreases monotonically 
in magnitude in the positive ^-direction; the density and pressure increase monotonically in 
that direction. For t > 2cJ(y—l)a, the inequality (94.10) does not hold for the piston 
velocity, and so the gas can no longer follow the piston. A vacuum is then formed in a region 
adjoining the piston, beyond which the gas velocity decreases from — 2c /(y— 1) to zero 
according to formula (1). 




If the piston moves into the pipe (U = at), a simple compression wave is formed; the 
corresponding solution is obtained by merely changing the sign of a in (1) (Fig. 65b). It is 
valid, however, only until a shock wave is formed ; the time when this happens is determined 
from formula (94.15), and is 

t = 2c Q ja{y+\). 



Problem 2. The same as Problem 1, but for the case where the piston moves in any 
manner. 

Solution. Let the piston begin to move at time t = according to the law x — X(t) 
(with X(0) = 0); its velocity is U = X'{t). The boundary condition on the piston (v = U 
for x = X) gives v = X'(t),f(v) = X(t)-t[c +Uy + l)X'it)]. If we now regard t as a para- 
meter, these two equations determine the function f(v) in parametric form. Denoting the 
parameter by t, we can write the solution as 



v = X\r), x = X{T) + {t-T)[c +\{y+\)X\T)l 



(1) 



which determines, in parametric form, the required function v(t, x) in the simple wave which 
is caused by the motion of the piston. 

Problem 3. Determine the time and place of formation of the shock wave when the piston 
(Problem 1) moves according to the law U = at n (m > 0). 

Solution. If a < 0, i.e. the piston moves out of the pipe, a simple rarefaction wave 
results, in which no shock wave is formed. We therefore assume that a > 0, i.e. the piston 
moves into the pipe, causing a simple compression wave. 



372 One-dimensional Gas Flow §95 

When the function v(x, t) is given by the parametric formulae (1) (Problem 2), and 
X = aT n+1 jn + l, the time and place of formation of the shock wave are given by the 
equations 

(8x \ 
— I = -CO + |^-lfl W (y+l)-lfl T «[ y -l +W ( y+ l)] = 0, 

/ 8 2 x \ ' ' 

i-^J = itTn-2an(n-l)(y+l)-^anrn-i[y-l+n(y+l)] = 0, 

where the second equation must be replaced by t = if we are concerned with the formation 
of a shock wave at the forward front of the simple wave. 

For n = 1 we find t = 0, t = 2c /a(y + l), i.e. the shock wave is formed at the forward 
front at a finite time after the motion begins, in accordance with the results of Problem 1. 

For n < 1, the derivative 8x/8t is of varying sign (and therefore the function v(x) for given 
t is many-valued) for any t > 0. This means that a shock wave is formed at the piston as 
soon as it begins to move. 

For n > 1 the shock wave is formed, not at the forward front of the simple wave, but at 
some intermediate point given by (1). Having determined r and t from (1), we can then 
find the place of formation of the discontinuity from (1) of Problem 2. The result is 



\ a I y+lln-r J 



„ /2co\ 1/re r y n-ll 1 

* = 2cq\ 1 — . 

\ a J Ly+1 »+ 1 J (n-l)<»-i>/»[y-l + n(y +l)]i/» 



§95. Formation of discontinuities in a sound wave 

A travelling plane sound wave, being an exact solution of the equations of 
motion, is also a simple wave. We can use the general results obtained in §94 
to derive some properties of sound waves of small amplitude in the second 
approximation (the first approximation being that which gives the ordinary 
linear wave equation). 

We must notice first of all that a discontinuity must ultimately appear in 
each wavelength of a sound wave. This leads to a very marked damping of 
the wave, as shown in §94. It must be remarked, however, that this happens 
only for a sufficiently strong sound wave ; a weak sound wave is damped by 
the usual effects of viscosity and thermal conduction before the effects of 
higher order in the amplitude can develop. 

The distortion of the wave profile has another effect also. If the wave 
is purely harmonic at some instant, it ceases to be so at later instants, on 
account of the change in shape of the profile. The motion, however, remains 
periodic, with the same period as before. When the wave is expanded in a 
Fourier series, terms with frequencies nco (n being integral and co being the 
fundamental frequency) appear, as well as that with frequency co. Thus the 
distortion of the profile as the sound wave is propagated may be regarded as 
the appearance in it of higher harmonics in addition to the fundamental 
frequency. 



§95 Formation of discontinuities in a sound wave 373 

The velocity u of points in the wave profile (the wave being propagated in 
the positive ^-direction) is obtained, in the first approximation, by putting in 
(94.11) v = 0, i.e. u = cq, corresponding to the propagation of the wave 
with no change in its profile. In the next approximation we have 

u = co + p' du/dpo = co+(du/dp )povjco, 

or, using the expression (92.10) for the derivative dujdp, 

u = co + ao^, (95.1) 

where we have put for brevity 

a = (c*l2V*)(&Vldp)s. (95.2) 

For a perfect gas, a = ^(y+l), and formula (95.1) agrees with the exact 
formula (see (94.8)) for the velocity u. 

In the general case of arbitrary amplitude, the wave is no longer simple 
after the discontinuities have appeared. A wave of small amplitude, however, 
is still simple in the second approximation even when discontinuities are 
present. This can be seen as follows. The changes in velocity, pressure and 
specific volume in a shock wave are related by #2—^1 = VKP2— Pi)(Vi— V2)]. 
The change in the velocity v over a segment of the #-axis in a simple wave is 

v 2 -v! = J ^(-8V/dp)dp. 
Pi 
A simple calculation, using an expansion in series, shows that these two 
expressions differ only by terms of the third order (it must be borne in mind 
that the change in entropy at a discontinuity is of the third order of smallness, 
while in a simple wave the entropy is constant). Hence it follows that, as far 
as terms of the second order, a sound wave on either side of a discontinuity in 
it remains simple, and the appropriate boundary condition is satisfied at the 
discontinuity itself. In higher approximations this is no longer true, on 
account of the appearance of waves reflected from the surface of discontinuity. 
Let us now derive the condition which determines the location of the dis- 
continuities in a travelling sound wave (again in the second approximation). 
Let u be the velocity of the discontinuity relative to a fixed co-ordinate 
system, and v\, v% the velocities of the gases on each side of it. Then the 
condition that the mass flux is continuous is pi(^i — u) = ^2(^2 — u), whence 
u — {p\v\ — p2P2)\(pi — p%). As far as the second-order terms, this is equal to 
the derivative d(pv)ldp at the point where v is equal to \{vi + V2) : 

u = [d(pv)/dp] v = i(Vl+Vt) . 

Since, in a simple wave, d{pv)jdp = v + c, we have, by (95.1), 

u = co + i<x(^i+^2)- (95.3) 

From this we can obtain the following simple geometrical condition which 
determines the position of the shock wave. In Fig. 66 the curve shows the 
velocity profile corresponding to the simple wave ; let ae be the discontinuity. 

13 



374 One-dimensional Gas Flow §95 

The difference of the shaded areas abc and cde is the integral 

{x~xq)6v 

taken along the curve abcde. In the course of time, the wave profile moves ; 



v 2 




*0 

Fig. 66 



let us calculate the time derivative of the above integral. Since the velocity 
dx/dt of points in the wave profile is given by formula (95.1), and the velocity 
dxojdt of the discontinuity by (95.3), we have 



dt 



V* V% Vt 

J (x— xo)d?; = ol{ J vdv—%(vi+V2) J dv} = 0; 



in differentiating the integral, we must notice that, although the limits of 
integration v\ and V2 also vary with time, x—xo always vanishes at the limits, 
and so we need only differentiate the integrand. 

Thus the integral J* (x— #o)dz> remains constant in time. Since it is zero at 
the instant when the shock wave is formed (the points a and e then coin- 
ciding), it follows that we always have 



I (x— #o)dz> = 0. 



(95.4) 



abcde 



Geometrically this means that the areas abc and cde are equal, a condition 
which determines the position of the discontinuity. 

Let us consider a single one- dimensional compression pulse, in which a 
shock wave has already been formed, and ascertain how this shock will finally 
be damped. By so doing, we also find the law of damping of any plane shock 
wave after it has been propagated for a sufficiently long time. 

In the later stages of its propagation, a sound pulse containing a shock 
wave will have a triangular velocity profile. Let the profile be given at some 
instant (which we take as t = 0) by the triangle ABC (Fig. 67a). If the 
points in this profile move with the velocities (95.1), we obtain after time / 



§95 



Formation of discontinuities in a sound wave 



375 



a profile A'B'C (Fig. 67b). In reality, the discontinuity moves to E, and 
the actual profile will be A'DE. The areas DB'F and C'FE are equal, by 
(95.4), and therefore the area A'DE of the new profile is equal to the area 
ABC of the original profile. Let / be the length of the sound pulse at time t, 
and Av the velocity discontinuity in the shock wave. During time t> the point 
B moves a distance cutAvo relative to C; the tangent of the angle B'A'C is 
therefore A^ /(/o + cutAvo), and we obtain the condition of equal areas ABC 
and A'DE in the form 



whence 



IqAvq = Z 2 A^ /(/o + a/A^o), 



/ = /oVU + aA^/A)], 

Av = Avol^[l + oLAvotjlo]. 



(95.5) 



(a) 




Av n 




Fig. 67 



For t -> oo the intensity of the shock wave diminishes asymptotically with 
time as \\*Jt (or, what is the same thing, with distance as Ify/x). The 
total energy of a travelling sound pulse (per unit area of its front) is 



E = po j v 2 dx = Eol V[l + aA*; tj% 



(95.6) 



where Eo is the energy at time t = 0. For t -> oo the energy also tends to 
zero as Ij-y/t. 

If we have a spherical outgoing sound wave, any small section of it can be 
regarded as plane at sufficiently large distances r from the origin. The 
velocity of any point in the wave profile is then given by formula (95.1). If, 
however, we wish to use this formula to follow the motion of any point in the 
wave profile over long intervals of time, we must take into account the fact 
that the amplitude of a spherical wave falls off inversely as the distance r, 
even in the first approximation. This means that, at any given point in the 
profile, v is not constant, as it is for a plane wave, but decreases as 1/r. If 



376 One-dimensional Gas Flow §95 

vq is the value of v (for a given point in the profile) at a (large) distance yq, 
we can put v = vorofr. Thus the velocity u of points in the wave profile is 
u — co+(x.voro[r. The first term is the ordinary velocity of sound, and cor- 
responds to movement of the wave without change in the shape of the profile 
(apart from the general decrease of the amplitude as \fr). The second term 
results in a distortion of the profile. The amount 8r of this additional move- 
ment of points in the profile during a time t = (r — yq)Jc is obtained by multi- 
plying by drjCQ and integrating from yq to r ; this gives 

8y = (ccvoYolco)log(YJYo). (95.7) 

Thus the distortion of the profile of a spherical wave increases as the logarithm 
of the distance, i.e. much more slowly than for a plane wave, where the dis- 
tortion Sx increases as the distance x traversed by the wave. 




Fig. 68 



The distortion of the profile ultimately leads to the formation of dis- 
continuities in it. Let us consider shock waves formed in a single spherical 
sound pulse which has reached a large distance from the source (the origin). 
The spherical case is distinguished from the plane case primarily by the fact 
that the region of compression must be followed by a region of rarefaction ; 
the excess pressure and the velocity of the gas particles in the wave must both 
change sign (see §69). The distortion of the profile results ultimately in the 
formation of two shock waves: one in the region of compression, and the 
other in the region of rarefaction (Fig. 68). f In the leading shock wave, the 
pressure increases discontinuously, then gradually decreases into a rarefac- 
tion, then again increases discontinuously in the second shock (but not to 
its unperturbed value, which is reached only asymptotically behind this 
shock). 

The manner of the final damping of the shock waves with time (or, what 
is the same thing, with the distance r from the source) is easily found in 
exactly the same way as for the plane case discussed above. Using the result 
(95.7), we find that, at sufficiently large distances, the thickness / of the sound 



f It should be mentioned that, since there is always ordinary damping (due to viscosity and 
thermal conduction) when sound is propagated in the gas, the slowness of the distortion in a spherical 
wave may have the result that it is damped before discontinuities can be formed. 



§95 



Formation of discontinuities in a sound wave 



377 



pulse (the distance between the two discontinuities) increases as log*(r/a), 
instead of as \/x for the plane case ; a is some constant length. The intensity 
of the leading shock wave is damped according to rAv ~ log~ ¥ (r/a), or 

A© ~ lfr \ogi(r/a). (95.8) 

Finally, let us consider the cylindrical case. The general decrease in the 
amplitude of an outgoing sound wave occurs in inverse proportion to yV, 
where r is the distance from the axis. Repeating the arguments given for the 
spherical case, we now find the velocity u of points in the wave profile to be 
u = co + a.vo\/(rolr), and so the displacement Sr of points in the profile, 
between ro and r is 



Sr = 2a(volco)\/ro(<\/r- ^/r ). 



(95.9) 




Fig. 69 



The cylindrical propagation of a compression pulse must be accompanied, 
as in the spherical case, by a rarefaction of the gas behind the compression. 
Two shock waves must therefore be formed in this case also. By the same 
method, we find the ultimate law of increase of the thickness of the pulse : 
/ ~ r% and the ultimate law of damping of the intensity of the shock wave : 
y/rAv ~ r _i , or 

Av ~ r-S. (95.10) 

The formation of discontinuities in a sound wave is an example of the 
spontaneous occurrence of shock waves in the absence of any singularity in 
the external conditions of the flow. It must be emphasised that, although a 
shock wave can appear spontaneously at a particular instant, it cannot dis- 
appear in the same manner. Once formed, a shock wave decays only asymp- 
totically as the time becomes infinite. 

PROBLEMS 

Problem 1. At the initial instant, the wave profile consists of an infinite series of "teeth", 
as shown in Fig. 69. Determine how the profile and energy of the wave change with time. 

Solution. It is evident that, at subsequent instants, the wave profile will be of the same 
form, with / unchanged but the height vt less than v . Let us consider one "tooth" : at time 
t — 0, the ordinate through the point where v — vt cuts off a part vtljv of the base of the 
triangle. During a time t, this point moves forward a distance ocvtf. The condition that the 
base of the triangle is unchanged in length is vtl lv +a.tvt = l , whence vt — *> /(l +ctv t/l ). 
As t -*■ oo, the wave amplitude diminishes as \jt. The energy is E = -E /(l +<zv t]l o y, i.e. it 
diminishes as ljt 2 for t -> co. 



378 One-dimensional Gas Flow §96 

Problem 2. Determine the intensity of the second harmonic formed by the distortion of 
the profile of a monochromatic spherical wave. 

Solution. Writing the wave in the form rv — A cos(kr—tot), we can allow for the distor- 
tion, in the first approximation, by adding Sr to r on the right-hand side of this equation, and 
expanding in powers of Sr. This gives, by (95.7), 

rv — A cos(kr—cQt) — (oLk/2co)A 2 log(r/ro) sin 2(kr— cot); 

here r must be taken as a distance at which the wave can still be regarded, with sufficient 
accuracy, as strictly monochromatic. The second term in this formula is the second harmonic 
in the spectral resolution of the wave. Its total (time average) intensity I % is 

h = (aP&l&rcoSpo) lo g 2(r/r )/i 2 , 

where 1^ = IttCqPqA 2 is the intensity of the first harmonic. 

§96. Characteristics 

The definition of characteristics, given in §79, as lines along which small 
disturbances are propagated (in the approximation of geometrical acoustics) is 
of general validity, and is not restricted to the plane steady supersonic flow 
discussed in §79. 

For one-dimensional non-steady flow, we can introduce the characteristics 
as lines in the art-plane whose slope dxjdt is equal to the velocity of propaga- 
tion of small disturbances relative to a fixed co-ordinate system. Disturbances 
propagated relative to the gas with the velocity of sound, in the positive or 
negative ^-direction, move relative to the fixed co-ordinate system with 
velocity v±c. The differential equations of the two families of characteristics, 
which we shall call C + and C_, are accordingly 

(dx/dt)+ = v + c, (dx/dt)- = v-c. (96.1) 

Disturbances transmitted with the gas are propagated in the atf-plane along 
characteristics belonging to a third family Co, for which 

(d*/d*)o = v. (96.2) 

These are just the "streamlines" in the atf-plane; cf. the end of §79.f It 
should be emphasised that, for characteristics to exist, it is no longer necessary 
for the gas flow to be supersonic. The "directional" propagation of distur- 
bances, as evidenced by the characteristics, is here simply due to the causal 
relation between the motions at successive instants. 

As an example, let us consider the characteristics of a simple wave. For a 
wave propagated in the positive ^-direction we have, by (94.5), x = t(v + c) + 
+f(v). Differentiating this relation, we have 

dx = (v + c)dt + [t + tc'(v)+f\v)]dv. 

Along a characteristic C + , we have dx = {v + c)dt ; comparing the two equa- 
tions, we find that along such a characteristic [t + tc'(v)+f'(v)]dv = 0. The 



t The same equations (96.1) and (96.2) determine the characteristics for non-steady spherically 
symmetrical flow, if x is replaced by the radial co-ordinate r (the characteristics now being lines in 
the ri-plane). 



§96 



Characteristics 



379 



expression in brackets cannot vanish identically, and therefore dv = 0, i.e. 
v = constant. Thus we conclude that, along any characteristic C + , the velo- 
city is constant, and therefore so are all other quantities. The same property 
holds for the characteristics C- in a wave propagated to the left. We shall see 
in §97 that this is no accident, but is a mathematical consequence of the nature 
of simple waves. 

From this property of the characteristics C + for a simple wave, we can in 
turn conclude that they are a family of straight lines in the atf-plane; the 
velocity is constant along the lines x = t[v + c(v)] +f(v) (94.5). In particular, 
for a similarity rarefaction wave (a simple wave with f(v) = 0), these lines 
form a pencil through the origin in the art-plane. For this reason, a similarity 
simple wave is sometimes said to be centred. 




Fig. 70 



Fig. 70 shows the family of characteristics C + for the simple rarefaction 
wave formed when a piston moves out of a pipe with acceleration. It is a family 
of diverging straight lines, which begin from the curve x = X(t) giving the 
motion of the piston. To the right of the characteristic x = c$t lies a region 
of gas at rest, where the characteristics become parallel. 

Fig. 71 is a similar diagram for the simple compression wave formed when a 
piston moves into a pipe with acceleration. In this case the characteristics are 
converging straight lines, which eventually intersect. Since every charac- 
teristic has a constant value of v, their intersection shows that the function 
v(x, t) is many-valued, which is physically meaningless. This is the geo- 
metrical interpretation of the result obtained in §94: a simple compression 
wave cannot exist indefinitely, and a shock wave must be formed in it. The 
geometrical interpretation of the conditions (94.12), which determine the 
time and place of formation of the shock wave, is as follows. The intersecting 
family of rectilinear characteristics has an envelope, which, for a certain 
least value of t, has a cusp ; this gives the instant at which many-valuedness 
first occurs. Every point in the region between the two branches of the en- 
velope is on three characteristics C + . If the equations of the characteristics 



380 



One-dimensional Gas Flow 



§96 



are given in the parametric form x = x(v), t = t(v), the position of the cusp 
is given by equations (94.12).f 

We shall now indicate briefly how the physical definition, given above, of 
the characteristics as lines along which disturbances are propagated corre- 
sponds to the mathematical sense of the word in the theory of partial diff- 
erential equations. Let us consider a partial differential equation of the form 



dU 8U d 2 d> 

A— + IB — - + C— - + D = 0, 

dx 2 dx dt 8t 2 



(96.3) 



Envelope 




Fig. 71 



which is linear in the second derivatives; the coefficients A, B, C, D can be 
any functions, both of the independent variables x, t and of the unknown 
function ^ and its first derivatives. $ Equation (96.3) is of the elliptic type if 
B 2 — AC < everywhere, and of the hyperbolic type if B 2 — AC > 0. In 
the latter case, the equation 



Adt 2 -2Bdxdt+Cdx 2 = 0, 



(96.4) 



or 



dx/dt =[B± V(B 2 -AC)]/C, (96.5) 

determines two families of curves in the xt-plane, the characteristics (for a 
given solution <f>(x, t) of equation (96.3)). We may point out that, if the co- 
efficients A, B, C are functions only of x and t, then the characteristics are 
independent of the particular solution </>. 

Let a given flow correspond to some solution <f> — <f>o(x, t) of equation 
(96.3), and let a small perturbation <£i be applied to it. We assume that this 
perturbation satisfies the conditions for geometrical acoustics to be valid: 
it does not greatly affect the flow (<£i and its first derivatives are small), but 



f The particular case where the shock wave occurs at the boundary of the gas at rest corresponds 
to that where one branch of the envelope is part of the characteristic x = c t. 

J The velocity potential satisfies an equation of this form in one-dimensional non-steady flow. 



§97 Riemann invariants 381 

varies considerably over small distances (the second derivatives of <f>i are 
relatively large). Putting in equation (96.3) <f> = fo + fa, we then obtain 
for <f>\ the equation 

d^d>! dUi dtfa 

dx 2 dx dt dt 2 

with <j> = <f> in the coefficients A, B, C. Following the method used in 
changing from wave optics to geometrical optics, we write <f>i = ae^ t where 
the function iff (the eikonal) is large, and obtain 

^)Wi^ + C (^) 2 = 0. (96.6) 

\ 8x / dx dt \dtl 

The equation of ray propagation in geometrical acoustics is obtained by 
equating dxjdt to the group velocity: dx[dt = dcojdk, where k = tyjdx, 
co^-difjjdt. Differentiating the relation Ak?-2Bkco + Cco 2 = 0, we 
obtain dxjdt = (Ba)-Ak)fcCa>-Bk), and, eliminating k\oi by the same 
relation, we again arrive at equation (96.5). 

PROBLEM 

Find the equation of the second family of characteristics in a centred simple wave. 

Solution. In a centred simple wave propagated into gas at rest to the right of it, we have 
xjt = v+c = c +Ur+l)v. The characteristics C + form the pencil * = constant Xt. The 
characteristics C_, on the other hand, are determined by the equation 

dx 3 — y x 4 

— = V — C = 7 7^0- 

dt y+1 t y+l 

Integrating, we find 

2 y+l ft \<3-7>/<r+D 

x = -cot H -com — 1 , 

y— 1 y—\ \ to / 

where the constant of integration has been chosen so that the characteristic C_ passes through 
the point * = c t , t = 1 on the characteristic C+ (x = c t) which is the boundary between 
the simple wave and the region at rest. 

The "streamlines" in the xf-plane are given by the equation 

dx 2 



whence 



dx 2 / X \ 

dt y+l\t J 



2 y+l It \ 2/( t +1) 
x — -cot -\ -co^ol — 1 



1 / * ' 

-coto\ — 
1 \ to, 



y—\ y—\ \ to I 

§97. Riemann invariants 

An arbitrary small disturbance is in general propagated along all three 
characteristics (C + , C-, Co) leaving a given point in the atf-plane. However, 
an arbitrary disturbance can be separated into parts each of which is pro- 
pagated along only one characteristic. 



382 One-dimensional Gas Flow §97 

Let us first consider isentropic gas flow. We write the equation of con- 
tinuity and Euler's equation in the form 

dp dp dv 

~ + v— + pc*~ = 0, 

dt dx dx 

dv dv 1 dp 

— + v— + -— = 0; 

dt dx p dx 

in the equation of continuity we have replaced the derivatives of the density 
by those of the pressure, using the formulae 

dp 

dt \dp/ s 

Dividing the first equation by ± pc and adding it to the second, we obtain 

dv 1 dp (dv 1 dp 

— ± -+ — ± 

dt pc dt \ dx pc dx 

We now introduce as new unknown functions 



I dp \ dp 1 dp dp 1 dp 

~ \dp) s dt~ c 2 dt' dx ~ c 2 dx 



)(v±c) = 0. (97.1) 



J+ = v+jdplpc, J- = v-jdpl P c, (97.2) 

which are called Riemann invariants. It should be remembered that, in 
isentropic flow, p and c are definite functions of p, and the integrals on the 
right-hand sides are therefore definite functions. For a perfect gas 

J + = v + 2cl(y-l), J- = v-2cl(y-l). (97.3) 

In terms of these quantities, the equations of motion take the simple form 

Yd d 1 X d d 1 

fc+^d-^ ' [» + <"-w--°- < 97 - 4) 

The differential operators acting on J + and /_ are just the operators of 
differentiation along the characteristics C + and C- in the atf-plane. Thus we 
see that / + and /_ remain constant along each characteristic C + or C- re- 
spectively. We can also say that small perturbations of J + are propagated 
only along the characteristics C + , and those of/- only along C— 

In the general case of anisentropic flow, the equations (97.1) cannot be 
written in the form (97.4), since dpfpc is not a perfect differential. These 
equations, however, still permit the separation of perturbations propagated 
along characteristics of only one family. For such perturbations are those 
of the form Sv ± Spjpc, where 8v and Sp are arbitrary small perturbations 
of the velocity and pressure. In order to obtain a complete system of equa- 
tions of motion, the equations (97.1) must be supplemented by the adiabatic 
equation 

Yd dl 

[« + "d'- ' (97 - 5) 

which shows that perturbations 8s are propagated along the characteristics Co- 



§97 



Riemann invariants 



383 



An arbitrary small perturbation can always be separated into independent 
parts of the three kinds mentioned. 

A comparison with formula (94.4) shows that the Riemann invariants (97.2) 
are the quantities which, in simple waves, are constant throughout the region 
of the flow at all times : /- is constant in a simple wave propagated to the right, 
and J + in one travelling to the left. Mathematically, this is the fundamental 
property of simple waves, from which follows, in particular, the property 
mentioned in §96: one family of characteristics consists of straight lines. For 
example,l et the wave be propagated to the right. Each characteristic C + has 
a constant value of / + and, furthermore, a constant value of/-, which value is 
the same everywhere. Since both / + and /_ are constant, it follows that y 
and/» are constant (and therefore so are all the other quantities), and we obtain 
the property of the characteristics C + deduced in §96, which in turn shows 
that they are straight lines. 



1 Simple wave 




2 Constant flow 
Fig. 72 



If the flow in two adjoining regions of the art-plane is described by two 
analytically different solutions of the equations of motion, then the boundary 
between the regions is a characteristic. For this boundary is a discontinuity 
in the derivatives of some quantity, i.e. it is a weak discontinuity, and there- 
fore must necessarily coincide with some characteristic. 

The following property of simple waves is of great importance in the theory 
of one-dimensional isentropic flow. The flow in a region adjoining a region of 
constant flow (in which v = constant, p = constant) must be a simple wave. 

This statement is very easily proved. Let the region 1 in the atf-plane be 
bounded on the right by a region (2) of constant flow (Fig. 72). Both in- 
variants J + and/- are evidently constant in the latter region, and both families 
of characteristics are straight lines. The boundary between the two regions 
is a characteristic C + , and the lines C + in one region do not enter the other 
region. The characteristics C- pass continuously from one region to the other, 
and carry the constant value of /_ into region 1 from region 2. Thus J- is 
constant throughout region 1 also, so that the flow in the latter is a simple 



wave. 



384 One-dimensional Gas Flow §97 

The ability of characteristics to "transmit" constant values of certain 
quantities throws some light on the general problem of initial and boundary 
conditions for the equations of fluid dynamics. In particular cases of 
physical interest, there is usually no doubt about the choice of these condi- 
tions, which is dictated by physical considerations. In more complex cases, 
however, mathematical considerations based on the general properties of 
characteristics may be useful. 



1 quantity 2 quantities Iquantity 1 quantity 2 quantities 2 quantities 

Ia 




Fig. 73 



For definiteness, we shall discuss a one-dimensional isentropic gas flow. 
Mathematically, a problem of gas dynamics usually amounts to the deter- 
mination of two unknown functions (for instance, v and p) in a region of the 
atf-plane lying between two given curves {OA and OB in Fig. 73a), on which 
the boundary conditions are known. The problem is to find how many 
quantities can take given values on these curves. In this respect it is very 
important to know how each curve is situated relative to the directions (shown 
by arrows in Fig. 73) of the two characteristics C + and C_ leavingf each 
point of it. Two cases can occur: either both characteristics lie on the same 
side of the curve, or they do not. In Fig. 73 a, the curve OA belongs to the 
first case and the curve OB to the second. It is clear that, for a complete 
determination of the unknown functions in the region AOB, the values of 
two quantities must be given on the curve OA (e.g. the two invariants / + 
and /_), and those of only one quantity on OB. For the values of the second 
quantity are "transmitted" to the curve OB from the curve OA by the charac- 
teristics of the corresponding family, and therefore cannot be given arbi- 
trarily.! Similarly, Figs. 73b and c show cases where one and two quantities 
respectively are given on each bounding curve. 

It should also be mentioned that, if the bounding curve coincides with a 
characteristic, two independent quantities cannot be specified on it, since 



t In the art-plane, the characteristics leaving a given point are those which go in the direction of 
t increasing. 

J An example of this case may be given as an illustration : the gas flow when a piston moves into or 
out of an infinite pipe. Here we are concerned with finding a solution of the equations of gas dynamics 
in the region of the art-plane lying between two lines, the positive a>axis and the line x = X(t) which 
gives the movement of the piston (Figs. 70, 71). On the first line the values of two quantities are 
given (the initial conditions v = 0, p = p for t = 0), and on the second line those of one quantity 
(v = u, where u(t) is the velocity of the piston). 



§97 Riemann invariants 385 

their values are related by the condition that the corresponding Riemann 
invariant is constant. 

The problem of specifying boundary conditions for the general case of 
anisentropic flow can be discussed in an entirely similar manner. 

Finally, we may make the following remark. We have everywhere above 
spoken of the characteristics of one-dimensional flow as lines in the xt -plane. 
The characteristics can, however, also be defined in the plane of any two 
variables describing the flow. For example, we can consider the characteris- 
tics in the w-plane. For isentropic flow, the equations of these characteristics 
are given simply by / + = constant, J- = constant, with various constants 
on the right; we call these characteristics V + and T_. For a perfect gas these 
are, by (97.3), two families of parallel lines (Fig. 74). 




Fig. 74 



It should be noted that these characteristics are entirely determined by 
the properties of the gas, and do not depend on any particular solution of the 
equations of motion. This is because the equation of isentropic flow in the 
variables v, c is (as we shall see in §98) a linear second-order partial differen- 
tial equation with coefficients which depend only on the independent vari- 
ables. 

The characteristics in the xt and vc planes are transformations of one 
another involving the particular solution of the equations of motion. The 
transformation need not be one-to-one, however. In particular, only one 
characteristic in the w-plane corresponds to a given simple wave, and all the 
characteristics in the atf-plane are transformed into it. For a wave travelling 
to the right (e.g.), it is one of the characteristics T_; the characteristics C_ 
are transformed into the line T-, and the characteristics C + into its various 
points. 

PROBLEM 

Find the general solution of the equations of one-dimensional isentropic flow of a perfect 
gas with y = 3. 

Solution. For y = 3 we have J±= v±c, and equations (97.4) have the general integral 

x = (v + c)t+fi(v + c), 

x = (v — c)t+fz(v — c), 

where / x and/ 2 are arbitrary functions. These two equations implicitly determine the required 



3 86 One-dimensional Gas Flow 



§98 



functions v(x, t) and c(x, t), and therefore all other quantities. We may say that, in this case, 
the two quantities v±c are propagated independently as two simple waves which do not 
interact. 

§98. Arbitrary one-dimensional gas flow 

Let us now consider the general problem of arbitrary one-dimensional 
isentropic gas flow (without shock waves). We shall first show that this 
problem can be reduced to the solution of a linear differential equation. 

Any one-dimensional flow (i.e. a flow depending on only one spatial co-or- 
dinate) must be a potential flow, since any function v(x, t) can be written 
as a derivative: v(x, t) = dcf>{x, t)/dx. We can therefore use, as a first integral 
of Euler's equation, Bernoulli's equation (9.3): d<j>jdt+\v 2 + w = 0. From 
this, we find the differential 

8<f> dd> 

d<f> = —dx + — dt 

8x dt 

= vdx-(\v 2 + w)dt. 

Here the independent variables are x and t; we now change to the independent 
variables v and w. To do so, we use Legendre's transformation; putting 

d<j> = d(xv)-xdv-d[t(w + ±v2)] + td(w+lv 2 ) 
and replacing <f> by a new auxiliary function 

X = <l> — xv + t(w+%v 2 ), 
we obtain 

dx = -xdv + td(zo+%v 2 ) = tdw + (vt-x)dv> 

where x is regarded as a function of v and w. Comparing this relation with 
the equation d* = (d x ldw)dw + (d x ldv)dv, we have t = d x [dw, vt-x 
= dx/dv, or 

t = dx/dw, x = vdx/dw-dxldv. (98.1) 

If the function xfa, «>) is known, these formulae determine v and to as 
functions of the co-ordinate x and the time t. 

We now derive an equation for X - To do so, we start from the equation 
of continuity, which has not yet been used: 

dp 8 dp dp dv 

— + —{pv) = — + v— + p— = 0. 

dt dx K dt dx H dx 

We transform this equation to one in terms of the variables v, to. Writing 
the partial derivatives as Jacobians, we have 

d(p,x) d(t,p) d(t,v) 
h v f- p = 0. 

d(t,x) d(t,x) H d(t,x) 



§98 Arbitrary one-dimensional gas flow 387 

or, multiplying by d(t, x)jd(w, v), 

d(p,x) 8(t,p) d(t,v) 

1- V h p = U. 

d(w,v) d(w,v) d(zv,v) 

To expand these Jacobians we must use the following result. According to 
the equation of state of the gas, p is a function of any two other independent 
thermodynamic quantities; for example, we may regard p as a function of w 
and s. If s = constant, we have simply p = p(w), and the density is inde- 
pendent of v. Expanding the Jacobians, we therefore have 

dp 8x dp dt 8t 

dw dv dw dv dw 

Substituting here the expressions (98.1) for t and x, we obtain 

p dw \ dw dv 2 ) dw 2 

Ifs = constant, we have dw = dp/p, whence dwjdp = 1/p. We can therefore 
write dp/dro = (dp/d»(dp/dw) = p\c 2 . We finally have for x the equation 

c *x_*x + e L= (98 . 2) 

8w 2 dv 2 dw 

here the velocity of sound c is to be regarded as a function of w. The problem 
of integrating the non-linear equations of motion has thus been reduced 
to that of solving a linear equation. 

Let us apply this result to the case of a perfect gas. We have c 2 = (y- \)w, 
and the fundamental equation (98.2) becomes 

(y _ 1)K ^_£^ + ^ = 0. (98.3) 

v/ ; dw 2 dv 2 dw 

This equation has an elementary general integral if (3-y)/(y- 1) is an even 
integer : 

(3_ y )/( y _l) = 2«, or y = (3 + 2«)/(2«+l), w = 0,1,2,.... (98.4) 

This condition is satisfied by monatomic (y = I, n = 1) and diatomic 
(y = 7 1 n = 2) gases. Expressing y in terms of n, we can rewrite (98.3) as 

2 ^_f* + ^ = . (98.5) 



2n+l dw 2 dv 2 dw 

We denote by xn a function which satisfies this equation for a given n 
For the function xo we have 

d 2 X o d 2 X o d X o 
2w — H = U. 

dw 2 dv 2 dw 



388 One-dimensional Gas Flow §98 

Introducing in place of to the variable u = \/(2w), we obtain 

This is just the ordinary wave equation, whose general solution is 

Xo = fi(u+v)+f 2 (u-v), 
/i and/ 2 being arbitrary functions. Thus 

Xo = /i[V(2«>) +v] +f 2 [V(2to)-v]. (98.6) 

We shall now show that, if the function X n is known, the function X n+i 
can be obtained by differentiation. For, differentiating equation (98.5) with 
respect to w, we easily find 

2 & 2 /8 Xn \ 2n + 3 d / d Xn \ d 2 / d x 



^f-+^T7-f -TT^ =0. 



2n+l dw 2 \ dw J 2n+l dw \ dw 



I dv 2 \dw) 



Putting v = v'V[(2n+ 1)/(2« + 3)], we have for d Xn \dw the equation 

2n + 3 dw 2 \ dw J dw\dw) 8v' 2 \ dw / ' 
which is equation (98.5) for the function X n+\ (w, v'). Thus we conclude that 
( >\ d t ^ d I /2«+l\ 

Xn+1 («,« ) = — *,(»,,,) = — Xn (^y— )• (98.7) 

Using this formula n times and taking Xo from (98.6), we find that the 
general solution of equation (98.5) is 

Qn 
X = -^^^ 2 ( 2n+1 ^ +v 'i + MV[2(2n+l)w]-v]}, 

or 

3"- 1 f F 1 [V[2{2n+l)w]+v] + F 2 [V[2(2n+l)w]-v]) 

x = ^( ^ ~ J — i }' ( 98 - 8 ) 

where Fi and F 2 are again two arbitrary functions. 

If we express w in terms of the velocity of sound by w = c 2 l(y-l) 
= %{2n + l)c 2 , the solution (98.8) becomes 

* = ay ]H c + ^ti) + h-2^t))- < 98 - 9 ) 

The expressions c± v((2n+ 1) = c± £(y- l)v which are the arguments of the 
arbitrary functions are just the Riemann invariants (97.3), which are constant 
along the characteristics. 



§98 Arbitrary one-dimensional gas flow 389 

In applications it is often necessary to calculate the values of the function 
x{v, c) on a characteristic. The following formulaf is useful for this purpose : 

with ±vj(2n+l) = c + a (a being an arbitrary constant). 

Let us now ascertain the relation between the general solution just found 
and the solution of the equations of gas dynamics which describes a simple 
wave. The latter is distinguished by the property that in it v is a definite 
function of to: v = v(w), and therefore the Jacobian A = d(v, w)jd(x, t) 
vanishes identically. In transforming to the variables v and zo, however, we 
divided the equation of motion by this Jacobian, and the solution for which 
A = is therefore "lost". Thus a simple wave cannot be directly obtained 
from the general integral of the equations of motion, but is a special integral 
of these equations. 

To understand the nature of this special integral, we must observe that it 
can be obtained from the general integral by a certain passage to a limit, 
which is closely related to the physical significance of the characteristics as 
the paths of propagation of small disturbances. Let us suppose that the region 
of the vw-plane in which the function x(v, w) is not zero becomes a very narrow 
strip along a characteristic. The derivatives of x in the direction transverse 
to the characteristic then take a very wide range of values, since x diminishes 
very rapidly in that direction. Such solutions x( v -> w ) °f the equations of 
motion must exist. For, regarded as a perturbation in the ^zu-plane, they 
satisfy the conditions of geometrical acoustics, and are therefore non-zero 
along characteristics, as such perturbations must be. 

It is clear from the foregoing that, for such a function x, the time t — dx[8w 
will take an arbitrarily large range of values. The derivative of x along the 
characteristic, however, is finite. Along a characteristic (for instance, a 
characteristic V-) we have 

d/_ 1 dp da; 1 da? 

dv pc dtv dv c dv 



t It is most simply derived by using Cauchy's theorem in the theory of functions of a complex 
variable. For an arbitrary function F(c-{- u) we have 

/ d \»-i F(c + u) nm _j d \n~i F(c + u) 



= 2»-i[ — 



cdc l c \ dc* I c 

2mi J y/z{z-c*y 

where the integral is taken along a contour in the complex jar-plane which encloses the point z — c 2 . 
Putting now u = c-\-a and substituting in the integral ■s/z = 2£ — c, we obtain 

1 (n-l)l /^ + a) 



2"- 1 7mi J C n (C-c) n 

where the contour of integration encloses the point £ = c; again applying Cauchy's theorem, we 
have the result (98.10). 



390 One-dimensional Gas Flow §98 

The derivative of x with respect to v along a characteristic, which we denote 
by —f( v )> i s therefore 

^ = ^ + _^.^ = ^ + C ^L = _f( \ 

dv dv dw dv dv dw ~ 

Expressing the partial derivatives of x in terms of x and t by (98.1), we 
obtain the relation x = (v + c)t+f(v) y i.e. the equation (94.5) for a simple 
wave. The relation (94.4), which gives the relation between v and cina 
simple wave, is necessarily satisfied, since/- is constant along a characteristic 

r_. 

We have shown in §97 that, if the solution of the equations of motion 
reduces to constant flow in some part of the atf-plane, then there must be a 
simple wave in the adjoining regions. The motion described by the general 
solution (98.8) must therefore be separated from a region of constant flow (in 
particular, a region of gas at rest) by a simple wave. The boundary between 
the simple wave and the general solution, like any boundary between two 
analytically different solutions, is a characteristic. In solving particular 
problems, the value of the function x (w> v) on this boundary characteristic 
must be determined. 

The "joining" condition at the boundary between the simple wave and the 
general solution is obtained by substituting the expressions (98.1) for x and 
t in the equation of the simple wave x — (v±c)t+f(v); this gives 

— ± c^- +f(v) = 0. 
dv dw 

Moreover, in a simple wave (and therefore on the boundary characteristic), 
we have dv = ± dpfpc = ± dwfc, or ± c = dw\dv. Substituting this in the 
above condition, we obtain 



8 X d x dw d x 

— + T -7- +f(?) = — +f(v) = 0, 

dv dw av dv 



or, finally, 



X = - J>)d*, (98.11) 



which determines the required boundary value of x . In particular, if the 
simple wave has a centre at the origin, i.e. if f(v) s 0, then x = constant; 
since the function x is defined only to within an additive constant, we can 
without loss of generality take x — on the boundary characteristic. 

PROBLEMS 

Problem 1. Determine the resulting flow when a centred rarefaction wave is reflected 
from a solid wall. 

Solution. Let the rarefaction wave be formed at the point x = at time t = 0, and 
propagated in the positive ^-direction ; it reaches the wall after a time t = l/c , where / is 



§98 



Arbitrary one-dimensional gas flow 



391 



the distance to the wall. Fig. 75 shows the characteristics for the reflection of the wave. 
In regions 1 and 1' the gas is at rest; in region 3 it moves with a constant velocity v = — C/.f 
Region 2 is the incident rarefaction wave (with rectilinear characteristics C+), and region 5 
is the reflected wave (with rectilinear characteristics C_). Region 4 is the "region of inter- 
action", in which the solution is required ; the linear characteristics become curved on entering 
this region. The solution is entirely determined by the boundary conditions on the segments 
ab and ac. On ab (i.e. on the wall) we must have v = for x = I; by (98.1), we hence obtain 
the condition dx\dv = — / for v = 0. The boundary ac with the rarefaction wave is part of 
a characteristic C_, and we therefore have c— $(y — l)v = c—v/(2n + l) = constant; since, 
at the point a, v = and c — c , the constant is c . On this boundary x must be zero, so 
that we have the condition x — for c— v/(2n + l) = c . It is easily seen that a function of 
the form (98.9) which satisfies these conditions is 



/(z«+iw d \ n-i n I"/ V Y l n ) 



a) 



and this gives the required solution. 




Fig. 75 



The equation of the characteristic ac is (see §96, Problem) 

* = -(2n+ l)c t + 2(n + l)/(fc //) (2n+1)/2(n+1) . 

Its intersection with the characteristic Oc 

xjt = c -i{y+l)U = c -2(n+l)Ul{2n+l) 

determines the time at which the incident wave disappears : 

/(2»+l)»+l£ w 



U. = 



[(2n + iy -C/]»+i' 



In Fig. 75 it is assumed that U < 2c [(y+l); in the opposite case, the characteristic Oc 
is in the negative ar-direction (Fig. 76). The interaction of the incident and reflected waves 
then lasts for an infinite time (not, as in Fig. 75, for a finite time). 



f If the rarefaction wave is due to a piston which begins to move out of a'pipe at a constant velocity, 
then U is the velocity of the piston. 



392 



One-dimensional Gas Flow 



§99 



The function (1) also describes the interaction between two equal centred rarefaction waves 
which leave the points x = and x = 21 at time t = and are propagated towards each other ; 
this is evident from symmetry (Fig. 77). 




Fig. 76 




Fig. 77 



Problem 2. Derive the equation analogous to (98.3) for one-dimensional isothermal flow 
of a perfect gas. 

Solution. For isothermal flow, the heat function w in Bernoulli's equation is replaced by 
[X = J* dp/p = C T 2 j dp/p = ct 2 logp, 

where ct z = {dp\dp)T is the square of the isothermal velocity of sound. For a perfect gas 
ct = constant. Taking the quantity ju. (instead of w) as an independent variable, we obtain, 
by the same method as in the text, the following linear equation with constant coefficients : 

9 d2 X , d X d2 X n 
d/j? dfj, dv 2 

§99. The propagation of strong shock waves 

Let us consider the propagation of a spherical shock wave of great intensity 
resulting from a strong explosion, i.e. from the instantaneous release of a 



§99 The propagation of strong shock waves 393 

large quantity of energy (which we denote by E) in a small volume ; we sup- 
pose that the shock is propagated through a perfect gas (L. I. Sedov, 1946). 

We shall consider the wave at relatively small distances from the source, so 
that the amplitude is still large. These distances are, nevertheless, supposed 
large in comparison with the dimensions of the source; this enables us to 
assume that the energy E is generated at a single point (the origin). 

If the shock wave is strong, the pressure discontinuity in it is very large. 
We shall suppose that the pressure p% behind the discontinuity is so large, 
compared with the pressure p\ of the undisturbed gas in front of it, that 
PzlPi > (y+l)/(y— !)• This means that we can everywhere neglect pi in 
comparison with /% and the density ratio pzjpi is equal to its limiting value 
(y+l)/(y-l);see§85. 

Thus the gas flow pattern is entirely determined by two parameters: the 
initial gas density pi, and the quantity of energy E generated in the explosion. 
From these parameters and the two independent variables (the time t and 
the radial co-ordinate r), we can form only one dimensionless combination, 
which we write as 

i = r( P1 /^)i/5. (99.1) 

Consequently, we have a certain type of similarity flow. 

We can say, first of all, that the position of the shock wave itself at every 
instant must correspond to a certain constant value g o of the dimensionless 
quantity £. This gives at once the manner in which the shock wave moves 
with time; denoting by ro the distance of the shock from the origin, we have 

ro = &(£* 2 //>i) 1/5 . (99.2) 

From this we find the rate of propagation of the shock wave (relative to the 
undisturbed gas, i.e. relative to a fixed co-ordinate system): 

mi = drojdt = 2r jSt. (99.3) 

It diminishes with time as t~*. 

The gas pressure p2, the density p% and the velocity V2 = «i— «2 (relative 
to a fixed co-ordinate system) just behind the discontinuity can be expressed 
in terms of «i by means of the formulae derived in §85. According to (85.5) 
and (85.6),f we have 

v 2 = 2wi/(y+ 1), pz = pi(y + l)/(y- 1), p 2 = 2p 1 u 1 *j(y+ 1). (99.4) 

The density is constant in time, while V2 and p^ diminish as f~ s and t'* 
respectively. We may also note that the pressure p2 due to the shock increases 
with the total energy of the explosion as E*. 

Let us next determine the gas flow throughout the region behind the shock. 



t We here denote by aj and Ug the velocities of the shock wave, relative to the gas, given by for- 
mulae (85.6). 



394 One-dimensional Gas Flow §99 

Instead of the gas velocity v, the density p, and the pressure p, we introduce 
dimensionless variables v\ p', p\ defined by 

4 r y+1 8pi r 2 

-V, P = Pl Y —-p', p= * ^ -p'. (99.5) 



5(y+l) * ' * * y-V * 25(y+l) * 2 

The quantities z/, p' andp' can be functions only of the dimensionless variable 
£ . On the surface of discontinuity (i.e. for $ = ft) they must have the values 

V ' = p' = p' = 1 for | = ft. (99.6) 

The equations of centrally-symmetrical adiabatic gas flow are 

dv 8v 1 dp dp d{pv) 2pv 

1- v — = , — + + = 0, 

dt or p dr dt dx r 



(99.7) 



Id 8 \ p 

__ + *;_ log— = 0. 

\dt 8r p? 



The last equation is the equation of conservation of entropy, with the ex- 
pression (80.12) for the entropy of a perfect gas substituted. After substitut- 
ing (99.5), we obtain a set of ordinary differential equations for the functions 
v\ p and p'. The integration of these equations is facilitated by the fact that 
one integral can be obtained immediately, using the following arguments. 

The fact that we have neglected the pressure pi of the undisturbed gas 
means that we neglect the original energy of the gas in comparison with the 
energy E which it acquires as a result of the explosion. It is therefore clear 
that the total energy of the gas within the sphere bounded by the shock 
wave is constant and equal to E. Furthermore, since we have a similarity 
flow, it is evident that the energy of the gas inside any sphere of a smaller 
radius, which increases with time in such a way that f = any constant (not 
only ft), must remain constant; the radial velocity of points on this sphere 
is v n = 2r/5* (cf. (99.3)). 

It is easy to write down the equation which expresses the constancy of this 
energy. On the one hand, an amount of energy d*. 4nr 2 pv(zo+%v 2 ) leaves 
the sphere (whose area is 4nr 2 ) in time dt. On the other hand, the volume of 
the sphere is increased in that time by dt . v n . 47rr 2 , and the energy of the gas 
in this extra volume is d£ . 4rrr 2 pv n (e+%v 2 ). Equating the two expressions, 
putting € = plp(y— 1) and w = ye, and introducing the dimensionless func- 
tions by (99.5), we obtain 

p' (y+l-2a>' 2 

~ = ^ T-' (99 ' 8 > 

p Zyv — y— 1 

which is the required integral. It automatically satisfies the boundary 
condition (99.6) at the surface of discontinuity. 

When the integral (99.8) is known, the integration of the equations is 



§99 The propagation of strong shock waves 395 

elementary though laborious. The second and third equations (99.7) give 



dv' l y+l\dlog// 

dlog£ \ 2 /dlog£ 



(99.9) 
d / P'\ ^ 5(y + \)-4v' 

dlogi \° g p'r) 2v'-(y+l)' 

From these two equations we can express the derivatives d^'/d log £ and 
d log p'/dv', by means of (99.8), as functions of v' only, and then an integra- 
tion with the boundary conditions (99.6) gives 

\l) = * L TTy J [—y^-\ • 

V2yv'-y-l -]".[ 5(y+l)-2(3y-l)fl' l v *[ y+ 1 -2v' y* 

* - L-^r-J L t^ J [~v=r~\ • 

13y2_7 y +12 5(y-l) 3 

^1 = 77! 777^ — 77> v 2 = — - — — » »* = 



(3y-l)(2y+l)' 2y+l ' 2y+l' 

13y 2 -7y+12 1 



"4 SB 7^ 77^ 7777; 77» "5 



(2-y)(3y-l)(2y+l)' y-2' 

(99.10) 

Formulae (99.8) and (99.10) give the complete solution of the problem. The 
constant £o is determined by the condition 



E ~]( ! r + £i) M '* r > 



which states that the total energy of the gas is equal to the energy E of the 
explosion. In terms of the dimensionless quantities, this condition becomes 

Iq5 25( 3 2-1) J (^ V2 + ^W = 1. (99.11) 

For instance, for air (y = f) this constant is £o = 1*033. 

The ratios v\vi and pjpz as functions of rjr Q = f /£ are easily seen from the 
above formulae to tend to zero as rjro -> 0, in the manner 

vfa ~ r/r , plp2 ~ (r/rofKv-U; (99.12) 

the ratio of pressures pjpz, however, tends to a constant, and the ratio of 
temperatures therefore tends to infinity. 



396 



One-dimensional Gas Flow 



§100 



Fig. 78 shows the quantities vjv%, pjp2 and p/p2 as functions of r/ro for 
air (y = 14). The very rapid decrease of the density into the sphere is 
noticeable: almost all the gas is in a relatively thin layer behind the shock 
wave. This is, of course, due to the fact that the gas on the surface of greatest 
radius (ro) has a density six times the normal density, f 



1-0 




v/v 2 X J \ 


0'5 


p/p 2 y 


/__^y \ 






P/P2J 



0-5 
Fig. 78 



1-0 



§100. Shallow-water theory 

There is a remarkable analogy between gas flow and the flow in a gravita- 
tional field of an incompressible fluid with a free surface, when the depth of 
the fluid is small (compared with the characteristic dimensions of the problem, 
such as the dimensions of the irregularities on the bottom of the vessel). 
In this case the vertical component of the fluid velocity may be neglected 
in comparison with its velocity parallel to the surface, and the latter may be 
regarded as constant throughout the depth of the fluid. In this {hydraulic) 
approximation, the fluid can be regarded as a "two-dimensional" medium 
having a definite velocity v at each point and also characterised at each point 
by a quantity h, the depth of the fluid. 

The corresponding general equations of motion differ from those obtained 
in §13 only in that the changes in quantities during the motion need not be 
assumed small, as they were in §13 in discussing long gravity waves of small 
amplitude. Consequently, the second-order velocity terms in Euler's equa- 
tion must be retained. In particular, for one-dimensional flow in a channel, 



f The results of calculations for other values of y are given by L. I. Sedov, Similarity and Dimen- 
sional Methods in Mechanics, Chapter IV, §11, Cleaver-Hume Press, London 1959. The 
corresponding problem with cylindrical symmetry is also discussed. 



§100 Shallow-water theory 397 

depending only on one co-ordinate x (and on the time), the equations are 

8h 8{vh) 

— + -±J- = 0, 

8t dX (100.1) 
dv 8v 8h 
1- v — = — g — ; 

8t 8x 8x 

the depth h is here assumed constant across the channel. 

Long gravity waves are, in a general sense, small perturbations of the flow 
now under consideration. The results of §13 show that such perturbations 
are propagated relative to the fluid with a finite velocity, namely 

C = y/(gh). (100.2) 

This velocity here plays the part of the velocity of sound in gas dynamics. 
Just as in §79, we can conclude that, if the fluid moves with velocities v < c 
{streaming flow), the effect of the perturbations is propagated both upstream 
and downstream. If the fluid moves with velocities v > c {shooting flow), 
however, the effect of the perturbations is propagated only into certain regions 
downstream. 

The pressure/) (reckoned from the atmospheric pressure at the free surface) 
varies with depth in the fluid according to the hydrostatic law/) = pg{h—z), 
where z is the height above the bottom. It is useful to note that, if we intro- 
duce the quantities 

h 
p = P h, p = jpdz = \pgh* = gpfr, (100.3) 

o 

then equations (100.1) become 

85 8{vp) 8v dv 1 8b 

-£ + -^ =0, — + v— = - - ■/, (100.4) 

8t 8x 8t 8x p 8x 

which are formally identical with the equations of adiabatic flow of a perfect 
gas with y = 2 {p ~ p~ 2 ). This enables us to apply immediately to shallow- 
water theory all the results of gas dynamics for flow in the absence of shock 
waves. If shock waves are present, however, the results of shallow-water 
theory differ from those of perfect-gas dynamics. 

A "shock wave" in a fluid in a channel is a discontinuity in the fluid height 
h s and therefore in the fluid velocity v (what is called a hydraulic jump). 
The relations between the values of the quantities on the two sides of the 
discontinuity can be obtained from the conditions of continuity of the fluxes 
of mass and momentum. The mass flux density (per unit width of the 
channel) is/ = pvh. The momentum flux density is obtained by integrating 
p -f pv* over the depth of the channel, and is 



f (p+pv 2 )dz = \pgh 2 + pv 2 h. 



398 One-dimensional Gas Flow §100 

The conditions of continuity therefore give two equations: 

vifa = v 2 h 2 , (100.5) 

»i 2 Ai+ W = v 2 2 h 2 + ±gh 2 2 . (100.6) 

These give the relations between the four quantities vi, v 2 , hi, h 2 , two of which 
can be specified arbitrarily. Expressing the velocities vi and v 2 in terms of 
the heights h\ and h 2 , we obtain 

*>i 2 = &h 2 (hi+h 2 )lhi, v 2 2 = ighi(hi + h 2 )/h 2 . (100.7) 

The energy fluxes on the two sides of the discontinuity are not the same, and 
their difference is the amount of energy dissipated in the discontinuity per 
unit time. The energy flux density in the channel is 



n 
q = J ( - + frApvdz = ij(gh+v 2 ). 



Using (100.7), we find the difference to be 

qi-q 2 = gj(hi 2 + h 2 2 )(h 2 -hi)/4hih 2 . 

Let the fluid move through the discontinuity from side 1 to side 2. Then the 
fact that energy is dissipated means that qi — q 2 > 0, and we conclude that 

h 2 > h u (100.8) 

i.e. the fluid moves from the smaller to the greater height. We then can 
deduce from (100.7) that 

vi > ci = V(gh), v 2 < c 2 = VW, (100.9) 

in complete analogy to the results for shock waves in gas dynamics. The 
inequalities (100.9) could also be derived as the necessary conditions for the 
discontinuity to be stable, as in §84. 



CHAPTER XI 

THE INTERSECTION OF 
SURFACES OF DISCONTINUITY 

§101. Rarefaction waves 

The line of intersection of two shock waves is, mathematically, a singular 
line of two functions describing the gas flow. The vertex of an acute angle on 
the surface of a body past which the gas flows is always such a singular line. 
It is found that the gas flow near the singular line can be investigated in a 
general manner (L. Prandtl and T. Meyer, 1908). 

In considering the region near a small segment of the singular line, we may 
regard the latter as a straight line, which we take as the #-axis in a system of 
cylindrical co-ordinates r, </», z. Near the singular line, all quantities depend 
considerably on the angle </>, but their dependence on the co-ordinate r 
is only slight, and for sufficiently small r it can be neglected. The dependence 
on the co-ordinate z is also unimportant; the change in the flow pattern over 
a small segment of the singular line may be neglected. 

Thus we have to investigate a steady flow in which all quantities are func- 
tions of <f> only. The equation of conservation of entropy, v«grad$ = 0, 
gives v$ dy/d<£ = 0, whence $ = constant,f i.e. the flow is isentropic. In 
Euler's equation we can therefore replace grad pjp by grad to: (vgrad)v 
= — grad zv. In cylindrical co-ordinates, we have three equations : 

^^V_V_ VfdVf VfVj, l^da; dv z _ 

r d<f> r ' r d<f> r r d<f>' d<f> 

From the last of these we have v z = constant, and without loss of generality 
we can put v z = 0, regarding the flow as two-dimensional; this is simply a 
matter of suitably defining the velocity of the co-ordinate system along the 
2-axis. The first two equations can be written 

^ = d«v/<ty, (101.1) 

dv$ \ \ dp dw 

d(f> / p d<f> d<f> 

Substituting (101.1) in (101.2), we have 

dvf dv r dw 

V,f d4 + Vr d$ = ~ d0 ' 



t If »* = 0, we easily deduce from the equations of motion given below that v r — 0, v t ^ 0. Such 
a flow would correspond to the intersection of surfaces of tangential discontinuity (with a discontinuous 
velocity v t ), and is of no interest, since such discontinuities are unstable. 

399 



400 The Intersection of Surfaces of Discontinuity §101 

or, integrating, 

w + hiv^+Vr 2 ) = constant. (101.3) 

We may notice that equation (101.1) implies that curl v = 0, i.e. we have 
potential flow, as a result of which Bernoulli's equation (101.3) holds. 
Next, the equation of continuity, div(pv) = 0, gives 

Using (101.2), we obtain 



(?-)('-<)-* 




/4 A 





Fig. 79 



The derivative dp [dp, or more correctly (dp/d/>) s , is just the square of the 
velocity of sound. Thus 

(IHf 1 -?)-- ^ 

This equation can be satisfied in either of two ways. Firstly, we may have 
dv^jdcji+Vr = 0. Then, from (101.2), p — constant and p = constant, and 
from (101.3) we find that v 2 = v r 2 + v^ 2 = constant, i.e. the velocity is constant 
in magnitude. It is easy to see that in this case the velocity is constant in 
direction also. The angle x between the velocity and some given direction in 
the plane of the motion is (Fig. 79) 

x = <£ + tan-i(V^)- ( 10L6 ) 

Differentiating this expression with respect to <j> and using formulae (101.1) 
and (101.2), we easily obtain 

d x /d«£ = -(vrlpv^dpldc/,. (101.7) 

Since p = constant, it follows that x = constant. Thus, if the first factor 
in (101.5) is zero, we have the trivial solution of a uniform flow. 



*-- f ^ . (101.9) 



§101 Rarefaction waves 401 

Secondly, equation (101.5) can be satisfied by putting X — v^jc* = 0, 
i.e. Vf = ±c. The radial velocity is given by (101.3). Denoting the constant 
in that equation by wq, we find that 

«V = ±c, v r = ±\Z[2(w -w)-c 2 ]. 

In this solution, the velocity component v^ perpendicular to the radius 
vector is equal to the local velocity of sound at every point. The total velocity 
v = V(*V 2 + ^r 2 ) therefore exceeds that of sound. Both the magnitude and 
the direction of the velocity are different at different points. Since the velocity 
of sound cannot vanish, it is clear that the function v^,(<f>), which is continuous, 
must everywhere be + c, or else everywhere — c. By measuring the angle <f> 
in the appropriate direction, we can take v^ = c. We shall see below that the 
choice of the sign of v r follows from physical considerations, and that the 
plus sign must be taken. Thus 

«V = c, v r = V[2(m-w)-c*]. (101.8) 

From the equation of continuity (101.4) we have d<f> = — d(pv^)fpv r . Sub- 
stituting (101.8) and integrating, we have 

d(pc) 

p\/[2(zvo-w)-c 2 ] 

If the equation of state of the gas and the adiabatic equation are known (we 
recall that s is constant), this formula can be used to determine all quantities 
as functions of the angle <f>. Thus formulae (101.8) and (101.9) completely 
determine the gas flow. 

Let us now study in more detail the solution which we have obtained. 
First of all, we notice that the straight lines <f> = constant intersect the stream- 
lines at every point at the Mach angle (whose sine is v^jv = cfv), i.e. they 
are characteristics. Thus one family of characteristics (in the ry-plane) 
is a pencil of straight lines through the singular point, and has an important 
property in this case: all quantities are constant along each characteristic. 
In this respect the solution concerned plays the same part in the theory of 
steady two-dimensional flow as does the similarity flow discussed in §92 
in the theory of non-steady one-dimensional flow. We shall return to this 
point in §107. 

It is seen from (101.9) that (pc)' < 0, the prime denoting differentiation 
with respect to <f>. Putting (pc)' = pd(pc)jdp and noticing that the derivative 
d(pc)jdp is positive (see (92.9)), we find that p < 0, and therefore so are the 
derivatives p' = c 2 p and zv' = p'jp. Next, from the fact that w' is negative 
it follows that the velocity v = ■\/\2{v)q — w)] increases with <f>. Finally, 
from (101.7), x > 0. Thus we have 

dp/d<p < 0, dp/dcf> < 0, dv/dcf> > 0, d x /d<£ > 0. (101.10) 

In other words, when we go round the singular point in the direction of flow, 
the density and pressure decrease, while the magnitude of the velocity in- 
creases and its direction rotates in the direction of flow. 



402 The Intersection of Surfaces of Discontinuity §101 

The flow just described is often called a rarefaction wave, and we shall 
use this name in what follows. 

It is easy to see that a rarefaction wave cannot exist throughout the region 
surrounding the singular point. For, since v increases monotonically with <f>, 
a complete circuit round the origin (i.e. a change of <j> by 2tt) would give a 
value for v different from the initial one, which is impossible. For this reason, 
the actual pattern of flow round the singular line must be composed of a 
series of sectors separated by planes <j> = constant which are surfaces of 
discontinuity. In each of these regions we have either a rarefaction wave or a 
flow with constant velocity. The number and nature of these regions for 
various particular cases will be established in the following sections . Here we 
shall simply mention that the boundary between a rarefaction wave and a 
uniform flow must be a weak discontinuity: it cannot be a tangential 
discontinuity (of v r ), since the normal velocity component v^ = c does not 
vanish on it. Nor can it be a shock wave, since the normal velocity component 
v^ must be greater than the velocity of sound on one side of such a dis- 
continuity and smaller on the other side, whereas in our problem we always 
have Vq = c on one side of the boundary. 

An important conclusion can be drawn from the foregoing. Disturbances 
which cause weak discontinuities evidently leave the singular line (the #-axis) 
and are propagated away from it. This means that the weak discontinuities 
bounding the rarefaction wave must be ones which leave this line, i.e. the 
velocity component v r tangential to the weak discontinuity must be positive. 
This justifies the choice of the sign of v r made in (101.8). 

Let us now apply these formulae to a perfect gas. In such a gas 
w ~ c2 l(v~ 1)> while the equation of the Poisson adiabatic can be written 

p C -2/(y-D = constant, ^ c -2y/(r-i) - constant; (101.11) 

cf. (92.13). Using these formulae, we can put the integral (101.9) in the form 

fy+1 r dc 



—JS! 



y-\ J V(c* 2 -c 2 )' 
where c# is the critical velocity (see (80.14)). Hence 

rr>«s~ J- — 



<f> = / cos -1 h constant, 

Vy-1 c* 

or, if we measure <j> in such a way that the constant is zero, 

^ = c = c m cos V[(y- l)/(y + l)tf. (101.12) 

According to formula (101.8) we therefore have 

^ = y^r* sin V^- < 10U3 > 

Next, using the Poisson adiabatic equation in the form (101.11), we can find 



§101 Rarefaction waves 

the pressure as a function of the angle <f> : 

(Y-l 



p = p m cogyHr 



-i) /^. 

Vy+r 



Finally, we have for the angle x (101.6) 

-1 /y-l 

-cot / 

y+1 Vy + 

the angles x and <f> being measured from the same initial line. 



403 



(101.14) 



* - ^^(yyw^)' (i ° u5) 



0-9 

0-8 

0-7 

^' 0-6 

°1 0-5 

^- 0*4 








































































140° 
120° 
100° 

80° a 










scjv 






































p/p 


















0-3 
0-2 

0-1 




































sx 










oO 
40° 














































20 







4C 


f 


8( 


F 


12 

4 
IG. i 


iO° 

so 


« 


50° 


2C 


X>° 






Since we must have v r > 0, c> 0, the angle <f> in these formulae can vary 
only between and <^nax, where 



<?W = W[(y+l)/(y-l)]- 



(101.16) 



This means that the rarefaction wave can occupy a sector whose angle does 
not exceed <£max; for a diatomic gas (air, for example), this angle is 219-3°. 
When <j> varies from to <£max, the angle x varies from \n to <£ma X . Thus the 
direction of the velocity in the rarefaction wave can turn through an angle 
not exceeding <£max-ih- (= 129-3° for air). 

For <f> = ^max the pressure is zero. In other words, if the rarefaction wave 
occupies the maximum angle, the weak discontinuity on one side is a boundary 
with a vacuum, and is, of course, a streamline; we have v$ = c = 0, 
v r - v = V[(y+l)/(y-l)] c # = ^max, i.e. the velocity is radial and attains 
its limiting value © max (see §80). 

Fig. 80 shows graphs of pip*, cjv and x as functions of the angle <f> for 
air (y = 1-4). 



404 



The Intersection of Surfaces of Discontinuity 



§101 



It is useful to note the form of the curve in the ©a^-plane defined by 
formulae (101.12) and (101.13) (called the velocity hodograph). It is an arc of 
an epicycloid between circles of radii v = c # and v = c%-\Z[(y+l)l(y— 1)] 
= *>max (Fig. 81). 




Fig. 81 



PROBLEMS 

Problem 1 . Determine the form of the streamlines in a rarefaction wave. 

Solution. The equation of the streamlines for two-dimensional flow is, in polar co- 
ordinates, dr/vr = rd<£/w<£. Substituting (101.12) and (101.13) and integrating, we obtain 

r = r cos-(r+i)/(y-i)^/[(y-l)/( y+ l)]^ 

These streamlines form a family of similar curves concave toward the origin, which is the 
centre of similarity. 

Problem 2. Determine the maximum possible angle between the weak discontinuities 
bounding a rarefaction wave, for given values v u c x of the gas velocity and the velocity of 
sound at one discontinuity. 

Solution. The angle <j> corresponding to the first discontinuity is, by (101.12), 



<f>l = 



y+1 



C\ 



COS" 



y-1 c* 

The value of <f> 2 is <f>m&x, so that the angle required is 



y+1 



fc-* 1 = Jhri 



sin" 



-ifl 



The critical velocity c* is given in terms of v x and c x by Bernoulli's equation : 



ZOi + ivi 2 = 



c? 



>- 1 



+ w = 



7+1 
2(7-1)' 



The maximum possible angle through which the gas velocity can turn in a rarefaction wave 
is accordingly, by (101.15), the difference Xmax = x(«£i) — xifa)'- 



Xmax 



y+1 . , C\ . Ci 

— sin -1 sin -1 — . 

y— 1 c# vi 



§102 The intersection of shock waves 405 



As a function of v-Jc x , Xmax is greatest for v^Cx = 1 : 

(y+1 
y-\ 



i i i y+1 1 

Xmax = fw / - - 1 



For v t /ci -> oo, Xmax tends to zero: 

Xmax 



2 Ci 



y— 1 V\ 



§102. The intersection of shock waves 

Shock waves can intersect along a line. In considering the flow near a small 
segment of this line, we can assume that it is a straight line, and that the sur- 
faces of discontinuity are planes. It is therefore sufficient to discuss the inter- 
section of plane shock waves. 

The line of intersection of two discontinuities is, mathematically, a singular 
line, as has already been mentioned at the beginning of §101. The flow 
pattern near this line consists of a number of sectors, in each of which we have 
either uniform flow or a rarefaction wave of the kind described in §101. 
It is possible to give a general classification of the possible types of intersec- 
tion of surfaces of discontinuity (L. Landau 1944). 

First of all, we must make the following remark. If the gas flow on both 
sides of a shock wave is supersonic, then (as mentioned at the beginning of 
§86) we can speak of the "direction" of the shock wave, and accordingly 
distinguish shock waves leaving the line of intersection from those reaching it. 
In the former case, the tangential velocity component is directed away from 
the line of intersection, and we can say that the disturbances which cause the 
discontinuity leave this line. In the latter case, the perturbations leave a 
point not on the line of intersection. 

If the flow on one side of the shock wave is subsonic, then disturbances are 
propagated in both directions along its surface, and the "direction" of the 
shock has, strictly, no meaning. In the arguments given below, however, 
what is important is that disturbances leaving the point of intersection can be 
propagated along such a discontinuity. In this sense, such shock waves play 
the same part in the following discussion as the purely supersonic shocks 
which leave the intersection, and we shall include both kinds in the term 
"shocks which leave the intersection". 

Figs. 82-86 show the flow patterns in a plane perpendicular to the line of 
intersection. We can assume, without loss of generality, that the flow occurs 
in this plane. The velocity component parallel to the line of intersection 
(which lies in all the planes of discontinuity) must be the same in all regions 
round the line of intersection, and can therefore be made to vanish by an 
appropriate choice of the co-ordinate system. 

It is easy to see that there can be no intersection of shock waves in which no 
shock reaches the intersection. For instance, in the intersection of two shock 
waves leaving the intersection, shown in Fig. 82a, the streamlines of the flow 
14 



406 The Intersection of Surfaces of Discontinuity §102 

incident from the left would deviate in opposite directions, whereas the 
velocity should be constant throughout region 2, and this difficulty cannot be 
overcome by adding any further discontinuities in region 2.f Similarly, we 
can see that the intersection of a shock wave and a rarefaction wave both 
leaving the intersection, shown in Fig. 82b, is impossible; although the 
velocity in region 2 can be constant in direction, the pressure cannot be con- 
stant, since it increases in a shock wave but decreases in a rarefaction wave. 





Shock Weak Tangential Streamline 

wave discontinuity discontinuity 

Fig. 82 



Next, since the intersection cannot affect shock waves reaching it, the 
simultaneous intersection (along a common line) of more than two such waves, 
which are due to other causes, would be an improbable coincidence. Thus 
only one or two shock waves can reach the intersection. 

The following fact is very important. The gas flowing past a point of 
intersection can pass through only one shock or rarefaction wave leaving this 
point. For example, let the gas pass through two successive shock waves 
leaving the point O, as shown in Fig. 82c. Since the normal velocity com- 
ponent V2n behind the shock Oa is less than C2, the velocity component in 
region 2 normal to the shock Ob must also be less than c%, in contradiction to 
a fundamental property of shock waves. Similarly, we can see that the gas 
cannot pass through two successive rarefaction waves, or a shock wave and a 
rarefaction wave, leaving the point O. 

These arguments evidently cannot be extended to shock waves reaching the 
point of intersection. 

We can now proceed to enumerate the possible types of intersection. Fig. 
83 shows an intersection involving one shock wave Oa reaching it and two 
shock waves Ob, Oc leaving it. This case may be regarded as the splitting of 
one shock wave into two. J It is easy to see that, besides the two shock waves 



t In order not to encumber the discussion with repetitive arguments, we shall not give similar 
considerations for cases where there are regions of subsonic flow and the shock leaving the inter- 
section is actually a shock wave bounded by a subsonic region. 

% It should be noticed that a shock wave cannot divide into a shock and a rarefaction wave; it is 
easily seen that the changes in the pressure and the direction of the velocities in the two waves leaving 
cannot be reconciled. 



§102 The intersection of shock waves 407 

leaving, there must be formed a tangential discontinuity Od lying between 
them, which separates the gas flowing through Ob from that flowing through 
Oc.f For the shock Oa is due to other causes, and is therefore completely 
defined. This means that the thermodynamic quantities (p and p, say) and 
the velocity v have given values in regions 1 and 2. There remain at our dis- 
posal, therefore, only two quantities (the angles giving the directions of the 
discontinuities Ob and Oc) with which to satisfy, in general, four conditions 
(the constancy of p, p and two velocity components) in the region 3-4, 
which would have to be satisfied in the absence of the tangential discon- 
tinuity Od. The addition of the latter, however, reduces the number of 
conditions to two (the constancy of the pressure and of the direction of the 
velocity). 




Fig. 83 



An arbitrary shock wave, however, cannot divide in this manner. A shock 
wave reaching the intersection is defined by two parameters (for a given 
thermodynamic state of gas 1), say the Mach number Mi of the incident 
stream and the ratio of pressures pi[p2- It can divide in two only in a certain 
region in the plane of these two parameters.^ 

Intersections involving two shock waves reaching them can be regarded as 
"collisions" of two shocks due to other causes. Here two essentially different 
cases are possible, as shown in Fig. 84. 

In the first case, the collision of two shock waves results in two other 
shock waves leaving the point of intersection. If all the necessary conditions 



t As usual, the tangential discontinuity in reality becomes a turbulent region. 

% The determination of this region involves very laborious algebraic calculations. The results that 
have been published (see, for example, R. Courant and K. O. Friedrichs, Supersonic Flow and 
Shock Waves, Interscience, New York 1948), unfortunately, are largely invalidated by the fact that 
they make no distinction between shock waves reaching and leaving the intersection. The ternary 
configurations therefore include also those where two shock waves reach the intersection and one 
leaves it. This, however, is the intersection of two shocks due to other causes, and therefore reaching 
the point of intersection with given values of all parameters. Their "fusion" into one shock is possible 
only when these arbitrary parameters are related in a certain way, and this would be an improbable 
coincidence. 



408 



The Intersection of Surfaces of Discontinuity 



§102 



are to be fulfilled, a tangential discontinuity must again be formed, and it 
must lie between the two resulting shock waves. 

In the second case, instead of two shock waves, there are formed one shock 
wave and one rarefaction wave. 




Fig. 84 



Two colliding shock waves are defined by three parameters (for instance, 
Mi and the ratios p\jp2, Pijpz)- The types of intersection just described are 
possible only for certain ranges of values of these parameters. If the values 
of the parameters do not lie in these regions, the collision of the shock waves 
must be preceded by their breaking up. 

Fig. 85 shows the reflection of a shock wave from the boundary between 
gas in motion and gas at rest. Region 5 contains gas at rest, separated from 
the gas in motion by a tangential discontinuity. In the two regions 1 and 4 
adjoining it, the pressure must be the same and equal top$. Since the pressure 
increases in a shock wave, it is clear that the shock wave must be reflected 
from the tangential discontinuity as a rarefaction wave 3, which reduces the 
pressure to its initial value. 

Finally, we may briefly discuss the intersection of a shock wave with a weak 
discontinuity arriving from an external source. Here two cases can occur, 



§102 



The intersection of shock waves 



409 



according as the flow behind the shock wave is supersonic or subsonic. In 
the former case (Fig. 86a), the weak discontinuity is "refracted" at the shock 
wave into the space behind the latter; the shock itself is not refracted at the 
intersection, but has a singularity of a higher order, like that at a weak dis- 
continuity. Moreover, the entropy change in the shock wave must cause 
behind it a "weak tangential discontinuity", at which the derivatives of the 
entropy are discontinuous. 




Fig. 85 





Weak 

tangential 

discontinuity 



Weak 
discontinuity 



Fig. 86 

If, however, the flow becomes subsonic behind the shock wave, the weak 
discontinuity cannot penetrate into this region, and it ceases at the point of 
intersection (Fig. 86b). The latter is now a singular point; it can be shown 
that the velocity distribution behind the shock wave has a logarithmic sin- 
gularity at this point. Furthermore, as in the previous case, a weak tangential 
discontinuity of the entropy must occur behind the shock wave.f 



t A detailed qualitative and quantitative analysis of the possible types of intersection of shock waves 
with weak discontinuities is given by S .P. D'yakov, Zhurnal experimental' noli teoreticheskoi fiziki 33 
948, 962, 1957; Soviet Physics JETP 6 (33), 729, 739, 1958; Doklady Akademii Nauk SSSR 99 
921, 1954. 



410 The Intersection of Surfaces of Discontinuity §103 

§103. The intersection of shock waves with a solid surface 

An important part in the phenomenon of steady intersection of shock waves 
with the surface of a body is played by their interaction with the boundary 
layer. This interaction is very complex, and has not yet been sufficiently 
investigated, either experimentally or theoretically. However, simple general 
arguments enable us to obtain some important results, which we shall now 
expound, f 

The pressure is discontinuous in a shock wave, and increases in the direc- 
tion of motion of the gas. Hence, if the shock wave intersects the surface, 
there must be a finite increment of pressure over a very short distance near 
the place of intersection, i.e. there must be a very large positive pressure 
gradient. We know, however, that such a rapid increase in pressure cannot 
occur near a solid wall (see the end of §40) ; it would cause separation, and the 
pattern of flow round the body is changed in such a way that the shock wave 
moves away to a sufficient distance from the surface. 

These arguments, however, do not apply when the shock wave is weak. 
It is clear from the proof given at the end of §40 that the impossibility of 
a positive pressure discontinuity at the boundary layer is a consequence of 
the assumption that this discontinuity is large: it must exceed a certain 
limit depending on the value of R, which diminishes when R increases. J 

Thus we reach the following important conclusions. The steady inter- 
section of strong shock waves with a solid surface is impossible. A solid 
surface can intersect only weak shock waves, and the limiting intensity is the 
smaller, the greater R. The maximum permissible intensity of the shock wave 
also depends on whether the boundary layer is laminar or turbulent. If the 
boundary layer is turbulent, the onset of separation is retarded (§45). In a 
turbulent boundary layer, therefore, stronger shock waves can leave the 
surface of the body than in a laminar boundary layer. ff 

To avoid misunderstanding, it should be emphasised that these arguments 
rely on the fact that the boundary layer exists in front of the shock wave 
(i.e. upstream of it). The results obtained therefore relate, in particular, to 
shock waves which leave the trailing edge, but not to those which leave the 
leading edge, of the body; the latter can occur, for instance in flow past an 
acute-angled wedge, a case which is discussed in detail in §104. In the latter 
case the gas reaches the vertex of the angle from outside, i.e. from a region 
in which there is no boundary layer. It is therefore clear that the present 



f The boundary layer necessarily contains a subsonic part adjoining the surface, into which the 
shock wave cannot penetrate. In speaking of the intersection, we ignore this fact, which does not affect 
the following discussion. 

% In §40, Problem, we have determined the smallest pressure change Ap over a distance Ax which 
can cause separation in a laminar boundary layer. In the present application, we are concerned with 
the pressure change over a distance of the order of the thickness 8 of the boundary layer, and obtain 
the following law governing the decrease of Ap when the Reynolds number increases : 

*PlP ~ l/R** ~ l/R# f . 

ft The existing published data do not enable us to specify the maximum permissible intensity. 



§103 The intersection of shock waves with a solid surface 411 

arguments do not deny that shock waves can occur which leave the vertex of 
such an angle. 

In subsonic flow, separation can occur only when the pressure in the main 
stream increases downstream along the surface. In supersonic flow, however, 
it is found that separation can occur even when the pressure decreases down- 
stream. Such a phenomenon can occur by the combination of a weak shock 
wave with a separation, the pressure increase necessary for separation taking 
place in the shock wave ; the pressure may either increase or decrease down- 
stream in the region in front of the shock wave. 




Fig. 87 

The data at present existing do not enable us to give a detailed picture of the 
complex phenomena involved in the "reflection" of a shock wave from the 
subsonic part of a boundary layer (or from the turbulent region beyond the 
line of separation). An important part in these phenomena must be played 
by the fact that the disturbances due to the shock wave can be propagated 
both upstream and downstream through the subsonic part of the boundary 
layer, and can cause further discontinuities in it. In particular, the formation 
of another weak shock wave upstream may result in separation, which "dis- 
places" a strong shock wave incident on the surface from outside. In Fig. 
87, the line a is the incident shock wave, and b the shock wave formed up- 
stream, which causes separation at the point O. When the incident shock is 
"reflected" from the subsonic part of the turbulent region, we should expect, 
in particular, that a rarefaction wave would be formed. 

All the above discussion relates only to a steady intersection, with the shock 
wave and the body at relative rest. Let us now consider non-steady intersec- 
tions, when a moving shock wave is incident on a solid body, so that the line 
of intersection moves on the surface. Such an intersection is accompanied by 
reflection of the shock wave: besides the incident wave, a reflected wave 
leaving the body is formed. 

We shall examine the phenomenon in a system of co-ordinates which moves 
with the line of intersection; in this system the shock waves are steady. 
The simplest type of reflection occurs when the reflected wave leaves the line 
of intersection itself; this is called regular reflection (Fig. 88). If the angle of 
incidence ai and the intensity of the incident shock are given, the flow in 
region 2 is uniquely determined. The gas velocity in the reflected shock must 



412 The Intersection of Surfaces of Discontinuity §103 

be turned through an angle such that it is again parallel to the surface. When 
this angle is given, the position and intensity of the reflected shock are ob- 
tained from the equation of the shock polar. For a given angle, the shock 
polar determines two different shock waves, those of the weak and strong 
families (§86). Experimental results show that in fact the reflected shock 
always belongs to the weak family, and we shall assume this in what follows. 
It should be pointed out that, when the intensity of the incident shock tends 
to zero, the intensity of the reflected shock v then tends to zero also, and the 
angle of reflection <X2 tends to the angle of incidence oci, as we should expect in 
accordance with the acoustic approximation. In the limit ai -> 0, the 
reflected shock of the weak family passes continuously into the shock ob- 
tained when a shock wave is incident "frontally" (§93. Problem 1). 




3 

y77777777777J77777777777777777777. 
Fig. 88 



The mathematical calculations for regular reflection (in a perfect gas) offer 
no difficulty in principle, but the algebra is extremely laborious. Here we 
shall give only some of the results.^ 

It is clear from the general properties of the shock polar that regular 
reflection is not possible for arbitrary values of the parameters of the incident 
wave (the angle of incidence ai and the ratio pi\p\). For a given ratio P2JP1 
there is a maximum possible angle ai&,J and for ai > <xifc regular reflection 
is impossible. As p^jpi -> oo, the maximum angle tends to sin -1 (l/y) 
(= 40° for air). As p2Jpi -»■ 1, «i& tends to 90°, i.e. regular reflection is 
possible for any angle of incidence. Fig. 89 shows ai^ as a function of Pijpz 
for air. 

The angle of reflection <*2 is not in general the same as the angle of incidence. 
There is a value a* of the angle of incidence such that, if ai < a # ,the angle 
of reflection <X2 < ai; if ai> a*, on the other hand, <*2 > ai. The value of 
a* is % cos -1 l(y— 1) (= 39-2° for air); it does not depend on the intensity 
of the incident wave. 



t A more detailed account of the reflection of shock waves is given by R. Courant and K. O. 
Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York 1948, and by W. Bleakney 
and A. H. Taub, Reviews of Modern Physics 21, 584, 1949. 

The solution of complex problems concerning the regular reflection of a shock wave at almost nor- 
mal incidence on the vertex of an angle close to 180°, and the diffraction of a shock wave at glancing 
incidence on the vertex of a similar angle, has been given by M. J. Lighthill (Proceedings of the 
Royal Society A198, 454, 1949; 200, 554, 1950). 

J This is the value of the angle of incidence for which the strong and weak reflected shocks 
coincide. 



§104 



Supersonic flow round an angle 



413 



For ai > auk regular reflection is impossible, and the incident shock wave 
must break up at a distance from the surface, so that we have the pattern shown 
in Fig. 90, with three shock waves, and a tangential discontinuity leaving the 
point where the incident shock wave divides. 



90" 






















80° 
70° 
60° 
* 50° 
40° 
30° 














































































































































20° 










































10° 










































1-0 0-9 0-8 0-7 0-6 0-5 0-4 0-3 0-2 0-1 


P\/P2 










Fi 


G. S 


9 











a» 




77777777777777777777777 
Fig. 90 



§104. Supersonic flow round an angle 

In investigating the flow near the vertex of an angle on the surface, it is 
again sufficient to consider small portions of the vertex and suppose it 
straight, the angle being formed by two intersecting planes. We shall speak 
of flow outside an angle if the angle is greater than 7r, and of flow inside an 
angle if it is less than it. 

Subsonic flow past an angle is not essentially different from the flow of an 
incompressible fluid. Supersonic flow, however, is entirely different; an 
important property of it is the occurrence of discontinuities leaving the vertex 
of the angle. 

Let us first consider the possible flow patterns when a supersonic gas 
stream reaches the vertex along one of the sides of the angle. In accordance 
with the general properties of supersonic flow, the stream remains uniform 
up to the vertex. The turning of the stream into the direction parallel to 



414 The Intersection of Surfaces of Discontinuity §104 

the other side of the angle occurs in a rarefaction wave leaving the vertex, 
and the flow pattern consists of three regions separated by weak discon- 
tinuities {Oa and Ob in Fig. 91): the uniform gas stream 1 moving along 
the side AO is turned into the rarefaction wave 2 and then moves, again 
with constant velocity, along the other side of the angle. It should be 
noticed that no turbulent region is formed; in a similar flow of an incom- 
pressible fluid, on the other hand, a turbulent region must be formed, with 
a line of separation at the vertex of the angle (Fig. 16, §35). 



0/> 

7777777777777777777, 



(a) 



0/ 

^77777777777777777^^ 



(b) 




Let v\ be the velocity of the incident stream (1 in Fig. 91), and c\ the 
velocity of sound in it. The position of the weak discontinuity Oa is deter- 
mined immediately from the Mach number Mi = v\\c\ by the condition 
that it intersects the streamlines at the Mach angle. The changes in velocity 
and pressure in the rarefaction wave are determined by formulae (101.12)- 
(101.15); all that is needed is the direction from which the angle </> in these 
formulae is to be measured. The straight line <j> = corresponds to 
v = c = c*; for Mi > 1, there is in fact no such line, since vjc > 1 every- 
where. However, if the rarefaction wave is imagined to be formally extended 
into the region to the left of Oa, we can use formula (101.12), and we find 



§104 Supersonic flow round an angle 415 

that the discontinuity Oa must correspond to a value of <f> given by 

ly+1 



A A +1 -l Cl 

<f>l = / rcos i— , 

Vy-1 c* 



and that <f> must increase from Oa to 03. The position of the discontinuity Ob 
is determined by the fact that the direction of the velocity becomes parallel to 
the side OB of the angle. 



/ 2 



1 



777777777777777777. 




1 
777777777777777777, 




Fig. 92 



Fig. 93 



The angle through which the stream turns in the rarefaction wave cannot 
exceed the value ^max determined in §101, Problem 2. If the angle /? round 
which the flow occurs is less than it— xmax> the rarefaction wave cannot turn 
the stream through the necessary angle, and we have the flow pattern shown 
in Fig. 91b. The rarefaction in the wave 2 then proceeds to zero pressure 
(reached on the line Ob\ so that the rarefaction wave is separated from the 
wall by a vacuum (region 3). 

The flow pattern described above is not the only possible one, however. 
Figs. 92 and 93 show patterns in which a region of gas at rest adjoins the 
second side of the angle, this region being separated from the moving gas by 
a tangential discontinuity; as usual, this becomes a turbulent region, so that 
the case considered corresponds to the presence of separation.f The stream 
is turned through a certain angle in a rarefaction wave (Fig. 92) or in a shock 
wave (Fig. 93). The latter case, however, is possible only if the shock wave is 
not too strong (in accordance with the general considerations given in §103). 

Which of these flow patterns will occur in any particular case depends in 
general on the conditions far from the angle. For instance, when gas flows 
out of a nozzle (the vertex of the angle being here the edge of the outlet), the 
relation between the pressure p\ of the outgoing gas and the pressure p e of 
the external medium is of importance. If p e < pi, the flow is of the type shown 
in Fig. 92; the position and angle of the rarefaction wave are then determined 



f According to experimental results, the compressibility of the gas somewhat diminishes the 
angle of the turbulent region resulting from the tangential discontinuity. 



416 The Intersection of Surf aces of Discontinuity §104 

by the condition that the pressure in regions 3 and 4 is equal to p e . The 
smaller p e , the greater the angle through which the stream must be turned. 
If, however, the angle /? (Fig. 92) is large, the gas pressure cannot reach the 
required value p e ; the direction of the velocity becomes parallel to the side 
OB of the angle before the pressure falls to p e . The flow near the edge of the 
outlet will then be as shown in Fig. 90. The pressure near the outer side 
OB of the outlet is entirely determined by the angle /5, and does not depend 
on the pressure p e \ the final decrease of the pressure to p e occurs only at a 
distance from the outlet. 




77777777777777-' " 
Fig. 94 



If p e > pi, on the other hand, the flow round the edge of the outlet is of 
the type shown in Fig. 93, with a shock wave which leaves the edge and raises 
the pressure from^i to p e . This is possible, however, only if the difference 
between p e and p\ is not too large, i.e. the shock wave is not too strong; 
otherwise there is separation at the inner surface of the nozzle, and the shock 
wave moves into the nozzle, in the manner described in §90. 

Next, let us consider flow inside an angle. In the subsonic case such a flow 
is accompanied by separation at a point ahead of the vertex (see the end of 
§40). For a supersonic incident flow, however, the change in direction may 
be effected by a shock wave leaving the vertex (Fig. 94). Here it must again 
be mentioned that such a simple separationless flow pattern is possible only 
if the shock wave is not too strong. Its intensity increases with the angle % 
through which the stream is turned, and we can therefore say that separation- 
less flow is possible only when % is not too large. 

Let us now consider the flow pattern which results when a free supersonic 
stream is incident on the vertex of an angle (Fig. 95). The stream is turned 
into directions parallel to the sides of the angle by shock waves leaving the 
vertex. As has been shown in §103, this is the exceptional case where a shock 
wave of arbitrary intensity can leave a solid surface. 

If we know the velocities vi and c\ in the incident stream, we can deter- 
mine the positions of the shock waves and the gas flow in the regions behind 
them. The direction of the velocity V2 must be parallel to the side OA of the 
angle : V2yfv2 X = tan x- Thus V2 and the angle <f> giving the position of the 
shock wave can be determined immediately from the shock polar, using a 
chord through the origin at the known angle x to the axis of abscissae (Fig. 
50), as explained in §86. We have seen that, for a given x> the shock polar 
gives two different shock waves, with different values of <f>. One of these 



§104 Supersonic flow round an angle 4YJ 

(corresponding to the point B in Fig. 50) is the weaker, and in general leaves 
the flow supersonic; the other, stronger, shock renders the flow subsonic. 
In the present case of flow past an angle on a finite solid surface, f we must 
always take the former, i.e. the weak shock. It should be borne in mind that 
this choice is really decided by the conditions of the flow far from the angle. 



Fig. 95 

In flow past a very acute angle (x small), the resulting shock wave must 
obviously be very weak. It is natural to suppose that, as the angle increases, 
the intensity of the shock increases monotonically ; this corresponds to a 
movement along the arc QC of the shock polar (Fig. 50), from Q towards C. 

We have also seen in §86 that the angle through which the velocity vector is 
turned in a shock wave cannot exceed a certain value Xmax, which depends 
on Mi. The flow pattern described above is therefore impossible if either of 
the sides of the angle makes an angle greater than xmax with the direction of 
the incident stream. In this case the gas flow near the angle must be sub- 
sonic; this is achieved by the appearance of a shock wave somewhere in front 
of the angle (see §114). Since xmax increases monotonically with Mi, we can 
also say that, for a given value of the angle x, Mi for the incident stream must 
be greater than a certain value Mi min . 

Finally it may be mentioned that, if the sides of the angle are situated, 
relative to the incident stream, as shown in Fig. 96, then a shock wave is of 
course formed on only one side of the angle ; the stream is turned on the other 
side by a rarefaction wave. 

PROBLEM 

Determine the position and intensity of the shock wave in flow past a very small angle 
(X <^ 1) for very large values of M : ( > 1/x). 



t The purely formal problem of flow past a wedge formed by the intersection of two infinite planes 
is of no physical interest. 



418 The Intersection of Surf aces of Discontinuity §105 

Solution. For x "^ 1 > the shock polar gives two values of <f>, one close to zero and the other 
close to \tt. The weak shock which we require corresponds to the former value, which is 
Kr+l)x; see §86, Problem 1. The ratio of pressures is, by (86.9), p^p x = JyCy+^MiV- 
The value of M behind the shock is 

1 

M 2 = - 



X \J y(y-l)' 
i.e. it is still large compared with unity, but not large compared with 1/x- 




Fig. 96 

§105. Flow past a conical obstacle 

The problem of steady supersonic flow near a pointed projection on the 
surface of a body is three-dimensional, and is very much more complicated 
than that of flow past an angle with a line vertex. No complete general in- 
vestigation of the former problem has yet been made. The only problem 
that has been completely solved is that of axially symmetric flow past a pro- 
jecting point, and we shall discuss this case. 

Near its vertex, an axially symmetric projection can be regarded as a right 
cone of circular cross-section, and so the problem consists in investigating the 
flow of a uniform stream past a cone whose axis is in the direction of inci- 
dence. The flow pattern is qualitatively as follows. 

As in the analogous problem of flow past a two-dimensional angle, a shock 
wave must be formed (A. Busemann 1929), and it is evident from symmetry 
that this shock is a conical surface coaxial with the cone and having the same 
vertex (Fig. 97 shows the cross-section of the cone by a plane through its 
axis). Unlike what happens in the two-dimensional case, however, the shock 
wave does not turn the gas velocity through the whole angle x necessary for 
the gas to flow along the surface of the cone (2^ being the vertical angle of the 
cone). After passing through the surface of discontinuity, the streamlines 
are curved, and asymptotically approach the generators of the cone. This 
curvature is accompanied by a continuous increase in density (besides the 
increase which occurs at the shock itself) and by a corresponding decrease in 
the velocity. Immediately behind the shock wave, the velocity is in general 
still supersonic (as in the two-dimensional case, it is determined by the 
"supersonic" part of the shock polar), but on the surface of the cone it 



§105 



Flow past a conical obstacle 



419 



may become subsonic. As in the two-dimensional case, for every value of 
the Mach number Mi = vijc\ for the incident stream, there is a limiting value 
Xmax for the angle of the cone, above which this type of flow becomes impos- 
sible. 

The conical shock wave intersects all streamlines in the incident flow at 
the same angle, and is therefore of constant intensity. Hence it follows (see 
§106) that we have isentropic potential flow behind the shock wave also. 




Fig. 97 



From the symmetry of the problem and its similarity properties (there are 
no characteristic constant lengths in the conditions imposed), it is evident 
that the distribution of all quantities (velocity, pressure) in the flow behind 
the shock wave will depend only on the angle 6 which the radius vector from 
the vertex of the cone to the point considered makes with the axis of the cone 
(the #-axis in Fig. 97). Accordingly, the equations of motion are ordinary 
differential equations; the boundary conditions on these equations at the 
shock wave are determined by the equation of the shock polar, while those 
at the surface of the cone are that the velocity should be parallel to the 
generators. These equations, however, cannot be integrated analytically, and 
have to be solved numerically. We refer the reader elsewheref for the results 
of the calculations, and merely give the curve (Fig. 51, §86) which shows the 
maximum possible angle xmax as a function of Mi. We may also mention 
that, as Mi -> 1, the angle xmax tends to zero: 



Xmax = constantx V[(Mi-l)/(y+l)], 



(105.1) 



t For example, N. E. Kochin, I. A. Kibel* and N. V. Roze, Theoretical Hydromechanics (Teoretich- 
eskaya gidromekhanika), Part 2, 3rd ed., p. 193, Moscow 1948; L. Howarth ed., Modern Developments 
in Fluid Dynamics: High Speed Flow, vol, 1, ch. 5, Oxford 1953. 



420 The Intersection of Surfaces of Discontinuity §105 

as may be deduced from the general law of transonic similarity (118.11); 
the constant is independent both of Mi and of the gas involved. 

An analytical solution of the problem of flow past a cone is possible only 
in the limit of small vertical angles. It is evident that in this case the gas 
velocity nowhere differs greatly from the velocity vi of the incident stream. 
Denoting by v the small difference between the gas velocity at the point 
considered and vi, and using its potential <f>, we can apply the linearised 
equation (106.4); if we take cylindrical co-ordinates x, r, co with the polar 
axis along the axis of the cone (co being the polar angle), this equation 
becomes 

or, for an axially symmetric solution, 

1 8 / 86 \ 8 2 6 

r 8r\ 8r / 8x 2 
where 

|5= V(* 2 -l)- (105.4) 

In order that the velocity distribution should be a function of 6 only, the 
potential must be of the form 6 — x f{£)> where g = rjx = tan 0. Sub- 
stituting this, we obtain for the function /(£) the equation 

£(l-/*W"+/ f = 0, 

of which the solution is elementary. The trivial solution / = constant 
corresponds to a uniform flow; the other solution is 

/ = constant x |y(l - PH 2 ) - cosh~i(l/^)]. 

The boundary condition on the surface of the cone (i.e. for f = tan x ~ x) 
is 

vrl{vi+v x ) « (l/©i)S0/0r = x (105.5) 

or/ ' = vix- Hence the constant is vix 2 , and we have the following expression 
r the potential in the region x > fir :f 

<f> = v lx *[y/{pP - £2 r 2) _ x co&h-i(x/pr)]. (105.6) 

It should be noticed that cf> has a logarithmic singularity for r -+ 0. 
We can now find the velocity components : 

v x = — vix 2 cosh -1 ( x I Br), 

K (105.7) 

v r = (vix 2 /r)\/(x 2 — 6 2 r 2 ). 

The pressure on the surface of the cone is calculated from formula (106.5); 



t In this approximation, the cone x = ]3r is a surface of weak discontinuity. 



§105 Flow past a conical obstacle 421 

since <j> has a logarithmic singularity for r -> 0, the velocity v r on the surface 
of the cone (i.e. for small r) is large compared with v x > and therefore we need 
retain only the term in v r 2 in the formula for the pressure. The result is 

P-Pi = />i^iV[log(2//?x)-i]. (105.8) 

All these formulae, which have been derived by means of a linearised theory, 
cease to be valid for large Mi, comparable with 1/x (see §119). 

The flow past a cone of arbitrary cross-section (the angle of attack being 
not necessarily zero) is a similarity flow, like the symmetrical flow past a 
circular cone. It has no characteristic length parameters, and so the velocity 
distribution can be a function only of the ratios yjx, zjx of the co-ordinates, 
i.e. it is constant along any straight line through the origin (the vertex of the 
cone). Such similarity flows are called conical flows.f 

PROBLEM 

Determine the flow past a cone of small vertical angle 2x placed at a small angle of attack a 
(C. Ferrari 1937)4 

Solution. We take the axis of the cone (not the direction of the main stream) as the #-axis ; 
the linearised equation (105.2) for the potential is unchanged if higher-order quantities 
('— < a<f>) are neglected, and the potential determines the gas velocity as V] +grad <f>. The boun- 
dary condition on the surface of the cone is 

v\ sin a cos co + v r 1 8<f> 

x a cos co -\ & x- 

v\ cosa+^s vi 8r 

We seek <f> as a sum : 

<f> = <p> (x, r) + cos co • <P (x, r), (1) 

where <£ (1 > is the expression (105.6), and <f> (2) satisfies the boundary condition 8<f> {i) l8r = — i^a. 
The function <f>^ can be written as rf(r/x) and, substituting r/cos to in equation (105.2), we 
obtain for / the equation 

f/"(^ 2 -l)+/'(2^ 2 -3) = 0. 

The trivial solution / = constant corresponds to a uniform stream incident (with velocity 
i> x a) in a direction perpendicular to the axis of the cone ; the other solution leads to 

<P = vtfxMWfrWix 2 - P 2 r 2 )- fir cosh-i(xlpr)]. 

The gas velocity is Vj+v'^+v* 2 ), where v (2 > = grad <f>W and v (1) is given by formulae 
(105 .7). The pressure is calculated from the formula 

p —pi = — Jpi{(^i cos a + 8<f>/dx) 2 + 
+ (vi sin a cos co + 8(f>Jdr) 2 + ( — v\ sin a sin co + 8</>/r 8co) 2 — v\ 2 } 

in whi ch the second-order terms in a and x must be retained. The pressure on the surface 
of the cone is found to be given by 

P-Pi = PiVi*{x 2 \og{2lp x )-\{ x 2 + «. 2 )- 

— lax cos w + a 2 cos 2co). 



f A detailed account of various problems concerning these flows is given by E. Carafoli, High 
Speed Aerodynamics {Compressible Flow), Pergamon Press, London 1958. 

J The solution of the same problem for any thin solid of revolution is given by F. I. Frankl' 
and E. A. Karpovich, Gas Dynamics of Thin Bodies, §2-7, Interscience, New York 1953. 



CHAPTER XII 

TWO-DIMENSIONAL GAS FLOW 

§106. Potential flow of a gas 

In what follows we shall meet with many important cases where the flow of 
a gas can be regarded as potential flow almost everywhere. Here we shall 
derive the general equations of potential flow and discuss the question of their 
validity. 

After passing through a shock wave, potential flow of a gas usually becomes 
rotational flow. An exception, however, is formed by cases where a steady 
potential flow passes through a shock wave whose intensity is constant over 
its area; such, for example, is the case where a uniform stream passes through 
a shock wave intersecting every streamline at the same angle, f The flow 
behind the shock wave is then potential flow also. To prove this, we use 
Euler's equation in the form 

|grada 2 -vxcurlv = -(1/p) grad/> 
(cf. (2.10)), or 

grad(w + %v 2 ) -vxcurlv= T grad s, 

where we have used the thermodynamic identity dzo = Tds + dp/p. In 
potential flow, however, w + %v 2 = constant in front of the shock wave, and 
this quantity is continuous at the shock; it is therefore constant everywhere 
behind the shock wave, so that 

vxcurlv = -Tgrads. (106.1) 

The potential flow in front of the shock wave is isentropic. In the general 
case of an arbitrary shock wave, for which the discontinuity of entropy varies 
over its surface, grad s # in the region behind the shock, and curl v is 
therefore also not zero. If, however, the shock wave is of constant intensity, 
then the discontinuity of entropy in it is constant, so that the flow behind the 
shock is also isentropic, i.e. grad s = 0. From this it follows that either 
curl v = or the vectors v and curl v are everywhere parallel. The latter, 
however, is impossible; at the shock wave, v always has a non-zero normal 
component, but the normal component of curl v is always zero (since it is 
given by the tangential derivatives of the tangential velocity components, 
which are continuous). 

Another important case where potential flow continues despite the shock 
wave is that of a weak shock. We have seen (§83) that in such a shock wave 



f We have already met with this situation in connection with supersonic flow past a wedge or 
cone (§§104, 105). 

422 



§106 Potential flow of a gas 423 

the discontinuity of entropy is of the third order relative to the discontinuity 
of pressure or velocity. We therefore see from (106.1) that curl v behind the 
shock is also of the third order. This enables us to assume that we have 
potential flow behind the shock wave, the error being of a high order of small- 
ness. 

We shall now derive the general equation for the velocity potential in an 
arbitrary steady potential flow of a gas. To do so, we eliminate the density 
from the equation of continuity div(/>v) = p div v+ vgrad p = 0, using Euler's 
equation 

(v-grad)v = -(l/p)grad/> = -(c 2 //>)grad/> 

and obtaining 

c 2 divv— v«(v«grad)v = 0. 

Introducing the velocity potential by v = grad <f> and expanding in components, 
we obtain the equation 

{c 2 -^)<f>x X + {c 2 -^)<j> y y+{c 2 -^)4> zz - 

f lUo.Z) 

— 2(<f> X <f>y(l> X y+<f>y<f> Z ff>y z +<f) z <l) X <f) zx ) — 0, 

where the suffixes here denote partial derivatives. In particular, for two- 
dimensional flow we have 

(c 2 - <f> X 2 )<f>XX + (c 2 - <f>y 2 )cf>yy ~ 2cf>x<f>y<f>xy = 0. (106.3) 

In these equations, the velocity of sound must itself be expressed in terms of 
the velocity ; this can in principle be done by means of Bernoulli's equation, 
to + \v 2 = constant, and the isentropic equation, s = constant. For a perfect 
gas, c as a function of v is given by formula (80.18). 

Equation (106.2) is much simplified if the gas velocity nowhere differs 
greatly in magnitude or direction from that of the stream incident from 
infinity.-)- This implies that the shock waves (if any) are weak, and so the 
potential flow is not destroyed. 

As in similar cases previously, we denote by v the small difference between 
the gas velocity at a given point and that of the main stream. Denoting the 
latter by vi, we therefore write the total velocity as vi + v. The potential 
<f> is taken to mean that of the velocity v: v = grad <f>. The equation for this 
potential is obtained from (106.2) by substituting (f> -xfy + xvi; we take the 
#-axis in the direction of the vector vi. We then regard ^ as a small quantity, 
and omit all terms of order higher than the first, obtaining the following 
linear equation: 

8 2 6 8 2 <t> 8 2 J> 

where Mi = vijci; the velocity of sound is, of course, given its value at 
infinity. 



t One such case was discussed in §105 (flow past a narrow cone), and others will be found in con- 
nection with gas flow past arbitrary thin bodies. 



424 Two-dimensional Gas Flow §106 

The pressure at any point is determined in terms of the velocity in the 
same approximation, by a formula which can be obtained as follows. We 
regard p as a function of w (for given s), and use the fact that {dw\dp) s = 1/p, 
writing p — pi « (dpldw) s (w — wi) = pi(w-w{). From Bernoulli's equation 
we have 

w-wi= _i[( Vl +v) 2 -vi 2 ] « -KV+^ 2 )-^i%, 
so that 

p-pi = -pivivx-ipi(v y 2 +v z 2 ). (106.5) 

In this expression the term in the squared transverse velocity must in general 
be retained, since, in the region near the #-axis (and, in particular, on the 
surface of the body itself), the derivatives 8<f>Jdy y dtfrjdz may be large com- 
pared with d<j>Jdx. 

Equation (106.4), however, is not valid if the number Mi is very close to 
unity {transonic flow), so that the coefficient of the first term is small. It is 
clear that, in this case, terms of higher order in the ^-derivatives of <£ must 
be retained. To derive the corresponding equation, we return to the original 
equation (106.2); when the terms which are certainly small are neglected, 
this becomes 

*<j>xx + <t>yy + 4>zz = 0. (106.6) 



(-S) 



In the present case, the velocity v x = v, and the velocity of sound c is 
close to the critical velocity c* (v now denoting the total velocity). We can 
therefore put c-c* = (v-c*) {dcjdv) v=Cif , or c-v = {c*-v)[\-(dcjdv) v = c ^. 
Using the fact that, for v = c = c^, we have by (80.4) dpjdv = —p]c, we 
put (for v = c*) 

dc dc dp p dc 

dv dp dv c dp 

so that 

c-v = [(c*-v)/c]d(pc)/dp = a*(c*-z>). (106.7) 

We have here used the expression (92.9) for the derivative d(pc)jdp, while a* 
denotes the value of a (95.2) for v = c*; for a perfect gas, a is constant, so 
that a* = a = l(y+ 1). To the same accuracy, this equation can be written 
as 

v/c-1 = a*(©/c*-l). (106.8) 

This gives the general relation between the Mach numbers M and M % in 
transonic flow. 

Using this formula, we can put 



^ 2 ~ v2 



2(1-^*2^(1-;) 



§107 Steady simple waves 425 

Finally, we introduce a new potential by the substitution <j> -+Cx(x + <j>), so 
that 

^ = ^_l, *_f* *_* (106.9) 

&c r* dy c* dz c* 

Substituting these formulae in (106.6), we obtain the following final equation 
for the velocity potential in a transonic flow (with the velocity everywhere 
almost parallel to the #-axis) : 

dd> cN> 8U dU 
2a*— -L = -L + -L. (106.10) 

8x dx 2 dy 2 dz 2 

The properties of the gas appear here only through the constant a*. We shall 
see later that this constant governs the entire dependence of the properties of 
transonic flow on the nature of the gas. 

The linearised equation (106.4) becomes invalid also in another limiting 
case, that of very large values of Mi : however, the appearance of strong shock 
waves has the result that potential flow cannot actually occur for such 
values of Mi (see §119). 

§107. Steady simple waves 

Let us determine the general form of those solutions of the equations of 
steady two-dimensional supersonic gas flow which describe flows in which 
there is a uniform plane-parallel stream at infinity, which then turns through 
an angle as it flows round a curved profile. We have already met a particular 
case of such a solution in discussing the flow near an angle ; the flow considered 
was essentially a plane-parallel one along one side of the angle, which turned 
at the vertex of the angle. In this particular solution all quantities (the two 
velocity components, the pressure and the density) were functions of only 
one variable, the angle <f>. Each of these quantities could therefore be expres- 
sed as a function of any other. Since this solution must be a particular case 
of the required general solution, it is natural to seek the latter on the assump- 
tion that each of the quantities p, p, v x , v y (the plane of the motion being 
taken as the ry-plane) can be expressed as a function of any other. This 
assumption is, of course, a very considerable restriction on the solution of 
the equations of motion, and the solution thus obtained is not the general 
integral of those equations. In the general case, each of the quantities p, 
P* Vx> Vy> which are functions of the two co-ordinates x, y, can be expressed 
as a function of any two of them. 

Since we have a uniform stream at infinity, in which all quantities, and in 
particular the entropy s, are constants, and since in steady flow of an ideal 
fluid the entropy is constant along the streamlines, it is clear that s = constant 
in all space if there are no shock waves in the gas, as we shall assume. 



426 Two-dimensional Gas Flow §107 

Euler's equations and the equation of continuity are 

dv x dv x 1 dp dv y 8v v 1 dp 

v x — \- Vy—— = , v x h v v = ; 

dx dy pdx dx V dy p dy' 

d d 

-rip®*) + —fay) = 0. 

dx dy 

Writing the partial derivatives as Jacobians, we can convert these equations 
to the form 

8 ( v x,y) 8(v x ,x) 1 d(p,y) 

V X Vy = , 

d{x,y) d(x,y) p d(x,y) 

d(v yy y) d(vy,x) 1 8(p,x) 

v x Vy = ; 

d(x,y) d{x,y) p d(x,y) 

d(pv x ,y) d{pv yi x) 
8(x,y) 8(x,y) 

We now take (say) x and p as independent variables. In order to effect this 
transformation, we need only multiply the above equations by d(x, y)jd{x, p), 
obtaining the same equations except that 8(x,p) replaces d(x,y) in the 
denominator of each Jacobian. We now expand the Jacobians, bearing in 
mind that all the quantities />, v x , v y are assumed to be functions of p but not 
of x, so that their partial derivatives with respect to x are zero. We then 
obtain 

/ ty\ dv x 1 dy / dy\ dvy 1 

r*-^/ # = - P Vx \ Vy - v *T x ) -& = - ? 

\ dx) dp ^\ dp dx dp) 

Here dy\dx denotes (dyjdx) p . All the quantities in these equations except 
dyjdx are functions of p only, by hypothesis, and x does not appear explicitly. 
We can therefore conclude, first of all, that dyjdx also is a function of p only: 
(dyjdx) p = fi{p), whence 

y = xfi(p)+f2(p), (107.1) 

where /2(/>) is an arbitrary function of the pressure. 

No further calculations are necessary if we use the particular solution, 
already known, for a rarefaction wave in flow past an angle (§§101, 104). 
It will be recalled that, in this solution, all quantities (including the pressure) 
are constants along any straight line (characteristic) through the vertex of 
the angle. This particular solution evidently corresponds to the case where the 
arbitrary function f2(p) in the general expression (107.1) is identically zero. 
The function f±(p) is determined by the formulae obtained in §101. 



§107 



Steady simple waves 



427 



Equation (107.1) for various constant/) gives a family of straight lines in 
the ry-plane. These lines intersect the streamlines at every point at the Mach 
angle. This is seen immediately from the fact that the lines y = xfi(p) in 
the particular solution with/ 2 = have this property. Thus one of the fami- 
lies of characteristics (those leaving the surface of the body) consists, in the 
general case, of straight lines along which all quantities remain constant; 
these lines, however, are no longer concurrent. 

The properties of the flow described above are, mathematically, entirely 
analogous to those of one-dimensional simple waves, in which one family of 
characteristics is a family of straight lines in the xt -plane (see §§94, 96, 97). 




'////V////////V//V/V////- 



Fig. 98 



Hence the class of flows under consideration occupies the same place in the 
theory of steady (supersonic) two-dimensional flow as do simple waves in 
non-steady one-dimensional flow. On account of this analogy, such flows 
are also called simple waves; in particular, the rarefaction wave which cor- 
responds to the case/2 = is called a centred simple wave. 

As in the non-steady case, one of the most important properties of steady 
simple waves is that the flow in any region of the #y-plane bounded by a 
region of uniform flow is a simple wave (cf. §97). 

We shall now show how the simple wave corresponding to flow round a 
given profile can be constructed. Fig. 98 shows the profile in question; 
to the left of the point O it is straight, but to the right it begins to curve. In 
supersonic flow the effect of the curvature is, of course, propagated only 
downstream of the characteristic OA which leaves the point O. Hence the flow 
to the left of this characteristic is uniform; we denote by the suffix 1 quanti- 
ties pertaining to this region. All the characteristics there are parallel and at 
an angle to the *-axis which is equal to the Mach angle a x = sin -1 (ci/*;i). 

In formulae (101.12)-(101.15), the angle <f> of the characteristics is measured 
from the line on which v = c = C*. This means (cf. §104) that the charac- 
teristic OA must have a value of <£ given by 

x /^ +1 -i Cl 

h = / — rcos 1— , 



428 Two-dimensional Gas Flow §107 

and the angle <f> is to be measured from OA' (Fig. 98). The angle between 
the characteristics and the #-axis is then 0*-^, where <f>* = ai + ^x. Accord- 
ing to formulae (101.12)-(101.15), the velocity and pressure are given in 
terms of (j> by 

v x = v cos 9, v y = v sin0, (107.2) 

* 2 = '4 1 + ^ sin V^]' (107.3) 

6 = ^-^tan-i(ygcoty^), (107.4) 

p = p* cos2r/(r-D j^—6. (107.5) 

V y+ 1 

The equation of the characteristics can be written 

y = xtan((f>*-cf>) + F(cf>). (107.6) 

The arbitrary function F(<f>) is determined as follows when the form of the 
profile is given. Let the latter be Y = Y(X), where X and Y are the co- 
ordinates of points on it. At the surface, the gas velocity is tangential, i.e. 

tan0 = dY/dX. (107.7) 

The equation of the line through the point (X, Y) at an angle <f> m -<f> to the 
x-axis is 

y-Y = {x-X) tan(<£ # -<£). 

This equation is the same as (107.6) if we put 

F(<f>) = Y-XtanO^-^). (107.8) 

Starting from the given equation Y = Y(X) and equation (107.7), we express 
the form of the profile in parametric equations X = X(6), Y = Y(6), the 
parameter being the inclination 6 of the tangent. Substituting 6 in terms of <f> 
from (107.4), we obtain X and Fas functions of <f>; finally, substituting these 
in (107.8), we obtain the required function F((f>). 

In flow past a convex surface, the angle 6 between the velocity vector 
and the x-axis decreases downstream (Fig. 98), and the angle <£*-0 between 
the characteristic and the x-axis therefore decreases monotonically also (we 
always mean the characteristic leaving the surface). For this reason, the 
characteristics do not intersect (in the region of flow, that is). Thus, in the 
region downstream of the characteristic OA (which is a weak discontinuity), 
we have a continuous (no shock waves) and increasingly rarefied flow. 

The situation is different in flow past a concave profile. Here the inclina- 
tion 6 of the tangent increases downstream, and therefore so does the inclina- 
tion of the characteristics. Consequently, the characteristics intersect in the 



§107 



Steady simple waves 



429 



region of flow. On different non-parallel characteristics, however, all 
quantities (velocity, pressure, etc.) have different values. Thus all these 
quantities become many-valued at points where characteristics intersect, 
which is physically impossible. We have already met a similar phenomenon 
in connection with a non-steady one-dimensional simple compression wave 
(§94). As in that case, it signifies that in reality a shock wave is formed. 
The position of the discontinuity cannot be completely determined from the 
solution under consideration, since this was derived on the assumption that 
there are no discontinuities. The only result that can be obtained is the 
place where the shock wave begins (the point O in Fig. 99, where the shock 
is shown by the continuous line OB). It is the point of intersection of charac- 
teristics whose streamline lies nearest to the surface of the body. On stream- 
lines passing below O (i.e. nearer to the surface) the solution is everywhere 




one-valued; its many-valuedness "begins" at O. The equations for the co- 
ordinates #o, yo of this point can be obtained in the same way as the cor- 
responding equations which determine the time and place of formation of 
the discontinuity in a one-dimensional non-steady simple wave. If we regard 
the inclination of the characteristics of a function of the co-ordinates (x, y) 
of points through which they pass, then this function becomes many-valued 
when x and y exceed certain values xq, yo. In §94 the situation was the same 
in relation to the function v(x, t), and so we need not repeat the arguments 
used there, but can write down immediately the equations 

{py\H)x = 0, {&y\m* = 0, (107.9) 

which now determine the place of formation of the shock wave. Mathemati- 
cally, this point is a cusp on the envelope of the family of straight charac- 
teristics (cf. §96). 

In flow past a concave profile, the simple wave exists along streamlines 
passing above O as far as the points where these lines intersect the shock 
wave. The streamlines passing below O do not intersect the shock wave at 
all, but we cannot conclude from this that the solution in question is valid 
at all points on these streamlines. The reason is that the shock wave has a 



430 Two-dimensional Gas Flow §108 

perturbing effect even on the gas which flows along these streamlines, and so 
alters the flow from what it would be in the absence of the shock wave. By 
a property of supersonic flow, however, these perturbations reach only the 
gas downstream of the characteristic OA (of the second family) which leaves 
the point where the shock wave begins. Thus the solution under considera- 
tion is valid everywhere to the left of AOB. The line OA itself is a weak 
discontinuity. We see that there cannot be a continuous (no shock waves) 
simple compression wave everywhere in flow past a concave surface, which 
would correspond to the simple rarefaction wave in flow past a convex surface. 
The shock wave formed in flow past a concave profile is an example of a 
shock which "begins" at a point inside the stream, away from the solid walls. 
The point where the shock begins has some general properties, which may be 
noted here. At the point itself the intensity of the shock wave is zero, and 
near the point it is small. In a weak shock wave, however, the discontinuities 
of entropy and vorticity are of the third order of smallness, and so the change 
in the flow on passing through the shock differs from a continuous potential 
isentropic change only by quantities of the third order. Hence it follows 
that, in the weak discontinuities which leave the point where the shock wave 
begins, only the third derivatives of the various quantities can be discontinu- 
ous. There will in general be two such discontinuities : a weak discontinuity 
coinciding with the characteristic, and a weak tangential discontinuity coin- 
ciding with the streamline (see the end of §89). 

§108. Chaplygin's equation: the general problem of steady two- 
dimensional gas flow 

Having dealt with steady simple waves, let us now consider the general 
problem of an arbitrary steady plane potential flow. We assume that the flow 
is isentropic and contains no shock waves. 

As was first shown by S. A. Chaplygin in 1902, it is possible to reduce 
this problem to the solution of a single linear partial differential equation. 
This is achieved by means of a transformation to new independent variables, 
the velocity components v x , v y ; this transformation is often called the hodo- 
graph transformation, the ^%-plane being called the hodograph plane and the 
xy-plane the physical plane. 

For potential flow we can replace Euler's equations by their first integral, 
Bernoulli's equation: 

w+&2 = w . (108.1) 

The equation of continuity is 

T&*) + TV*) = 0. (108.2) 

dx By 

For the differential of the velocity potential <f> we have d<j> = v x dx+v y dy. 
We transform from the independent variables x, y to the new variables v x , v y 



§108 Chaplygiris equation 431 

by Legendre's transformation, putting 

d<f> = d(xv x )-xdv x + d(yvy)-ydv y , 
introducing the function 

d> = -cff + xvz+yvy, (108.3) 

and obtaining 

d<I> = xdv x +ydvy, 

where <1> is regarded as a function of v x and v y . Hence 

x = d<bldv x , y = dQjdvy. (108.4) 

It is more convenient, however, to use, not the Cartesian components of the 
velocity, but its magnitude v and the angle 9 which it makes with the a:-axis : 

•v x = vcosd, Vy = vsin6. (108.5) 

The appropriate transformation of the derivatives gives, instead of (108.4), 

3<J> sin0 8® . S<D cos0 dd> /ino , x 
x = cos e- — , y = sin 0— - + — . (108.6) 

8v v dQ 8v v 86 

The relation between the potential <p and the function O is given by the 
simple formula 

<£ = -®+vd®ldv. (108.7) 

Finally, in order to obtain the equation which determines the function 
0(a, 0), we must transform the equation of continuity (108.2) to the new 
variables. Writing the derivatives as Jacob ians: 

d{pwx,y) d(pvy,x) _ 
8(x,y) d(x,y) 
multiplying by d(x } y)]d(v, 6) and substituting (108.5), we have 
8(pv cos 0, y) 8{pv sin 6, x) _ 
8(v,6) d(v, 0) 

To expand these Jacobians, we must substitute (108.6) for x and y. Further- 
more, since the entropy s is a given constant, if we express the density as a 
function of s and zv and substitute to = Wq - %v 2 we find that the density can 
be written as a function of v only: p = p(v). We therefore obtain, after a 
simple calculation, the equation 

d(pv) I 8® 1 S2<X> \ 32$ 

-Jl-L { + + p v — — = 0. 

dv \8v v 86 2 J dv 2 

According to (80.5), 

d(pv) / v* \ 



432 Two-dimensional Gas Flow §108 

and so we have finally Chaplygin's equation for the function <!>(#, 6) : 

6 2 <J> v 2 8 2 <& 3<X> 

+ 7-^5 -^- + ^^T = °- (108.8) 



£02 1-^2 &,2 ^ 

Here the velocity of sound is a known function c(v), determined by the 
equation of state of the gas together with Bernoulli's equation. 

The equation (108.8), together with the relations (108.6), is equivalent to 
the equations of motion. Thus the problem of solving the non-linear equa- 
tions of motion is reduced to the solution of a linear equation for the function 
®(v, 6). It is true that the boundary conditions on this equation are non- 
linear. These conditions are as follows. At the surface of the body, the gas 
velocity must be tangential. Expressing the equation of the surface in the 
parametric form X = X(6), Y = Y(0) (as in §107), and substituting X and 
Y in place of x and y in (108.6), we obtain two equations, which must be 
satisfied for all values of 0; this is not possible for every function ®(v, 6). 
The boundary condition is, in fact, that these two equations are compatible 
for all 6, i.e. one of them must be deducible from the other. 

The satisfying of the boundary conditions, however, does not ensure that 
the resulting solution of Chaplygin's equation determines a flow that is 
actually possible everywhere in the physical plane. The following condition 
must also be met: the Jacobian A = 8(x,y)jd(d, v) must nowhere be zero, 
except in the trivial case when all its four component derivatives vanish. 
It is easy to see that, unless this condition holds, the solution becomes com- 
plex when we pass through the line (called the limiting line) in the xy-plane 
given by the equation A = O.f For, let A = on the line v = v (6), and 
suppose that (dyjdd) v ^ 0. Then we have 

-a(-\ - 8(x,y) 8{v,d) - d ^ y) = ( 8x \ =0 

\8yJ v 8(v,d) d(v,y) 8{v,y) \8v! y 

Hence we see that, near the limiting line, v is determined as a function of x 
(for given y) by 

x — xq — %(d 2 xldv 2 ) y (v — vo) 2 , 

and v becomes complex on one side or the other of the limiting line. J 

It is easy to see that a limiting line can occur only in regions of supersonic 
flow. A direct calculation, using the relations (108.6) and equation (108.8), 
gives 



1 

A = - 
v 



e 2 d> l ao>\ 2 v 2 i » 
+ 



86 dv v 89 J \-v 2 \c 2 \ 8v 2 



(108.9) 



t There is no objection to a passage through points where A becomes infinite. If 1/A = on some 
line, this merely means that the correspondence between the xy and vQ planes is no longer one-to- 
one : in going round the xy-plane, we cover some part of the w#-plane two or three times. 

X This result clearly remains valid even if (8 2 xl8v 2 ) y vanishes with A but {dxjdv) y again changes 
sign for v = v , i.e. the difference x—x is proportional to a higher even power of v — vq. 



§109 Characteristics in steady two-dimensional flow 433 

It is clear that, for v ^ c, A > 0, and A can become zero only if v > c. 

The appearance of limiting lines in the solution of Chaplygin's equation 
indicates that, under the given conditions, a continuous flow throughout the 
region is impossible, and shock waves must occur. It should be emphasised, 
however, that the position of these shocks is not the same as that of the 
limiting lines. 

In §107 we discussed the particular case of steady two-dimensional super- 
sonic flow (a simple wave), which is characterised by the fact that the velocity 
in it is a function only of its direction: v — v(6). This solution cannot be 
obtained from Chaplygin's equation, since 1/A = 0, and the solution is 
"lost" when the equation of continuity is multiplied by the Jacob ian A in the 
transformation to the hodograph plane. The situation is exactly analogous 
to that found in the theory of non-steady one-dimensional flow. The re- 
marks made in §98 concerning the relation between the simple wave and the 
general integral of equation (98.2) are wholly applicable to the relation be- 
tween the steady simple wave and the general integral of Chaplygin's equation. 

The fact that the Jacobian A is positive in subsonic flow enables us to 
demonstrate an interesting theorem due to A. A. Nikol'skii and G. I. 
Taganov (1946). We have identically 

1 _ 8(0, v) _ 8(6, v) 8(x,v) 

A 8(x,y) 8(x,v) 8(x,y) 
or 

A \8x/ v \dyj 

In a subsonic flow A > 0, and we see that the derivatives (ddjdx) v and 
(dvjdy) x have the same sign. This has a simple geometrical significance : if 
we move along a line v = constant = vq, with the region v < vo to the right, 
the angle 6 increases monotonically, i.e. the velocity vector turns always 
counterclockwise. This result holds, in particular, for the line of transition 
between subsonic and supersonic flow, on which v = c = c%. 

In conclusion, we may give Chaplygin's equation for a perfect gas, writing 
c explicitly in terms of v : 

Q2® l- (y -l)„2 /(y+1K 2 £2$ g<J> 

+ v 2 V v = 0. (108.11) 



_ = | - ( _ . (108.10) 



86 2 \-v 2 \c* 2 8v 2 8v 

This equation has a family of particular integrals expressible in terms of 
hypergeometric functions.-]" 

§109. Characteristics in steady two-dimensional flow 

Some general properties of characteristics in steady (supersonic) two- 
dimensional flow have already been discussed in §79. We shall now derive 



t See, for instance, L. I. Sedov, Two-dimensional Problems of Hydrodynamics and Aerodynamics 
(Ploskie zadachi gidrodinamiki i aerodinamiki), Moscow 1950. 



434 Two-dimensional Gas Flow §109 

the equations which give the characteristics in terms of a given solution of the 
equations of motion. 

In steady two-dimensional supersonic flow there are, in general, three 
families of characteristics. All small disturbances, except those of entropy 
and vorticity, are propagated along two of these families (which we call the 
characteristics C + and C_); disturbances of entropy and vorticity are pro- 
pagated along characteristics (C ) of the third family, which coincide with 
the streamlines. For a given flow, the streamlines are known, and the problem 
is to determine the characteristics belonging to the first two families. 

The directions of the characteristics C + and C_ passing through each point 
in the plane lie on opposite sides of the streamline through that point, and make 
with it an angle equal to the local value of the Mach angle a (Fig. 41, §79). 
We denote by m the slope of the streamline at a given point, and by m +> m- 
the slopes of the characteristics C + , C_. Then we have 



whence 



m + — mo 
1 + motn + 


m-—niQ 

= tana, — = 

1 + mom- 
mo + tan a 
1 + mo tan a 


— tan a, 



the upper signs everywhere relate to C + and the lower to C_. Substituting 
m = Vyjv x , tana = cj^{v 2 -c 2 ) and simplifying, we obtain the following 
expression for the slopes of the characteristics : 

/ dy\ v x v y ± c\/(v 2 - c 2 ) 

If the velocity distribution is known, this is a differential equation which 
determines the characteristics C + and C-.f 

Besides the characteristics in the xy-plane, we may consider those in the 
hodograph plane, which are especially useful in the discussion of isentropic 
potential flow; we shall take this case in what follows. Mathematically, these 
are the characteristics of Chaplygin's equation (108.8), which is of hyperbolic 
type for v > c. Following the general method familiar in mathematical 
physics (see §96), we form from the coefficients the equation of the charac- 
teristics : 



dv* + dd 2 v 2 /(l-v 2 lc*) = 0, 



or 



/d0\ 1 \iv 2 \ 

Ur^Jh- 1 )- (m2) 



t Equation (109.1) also determines the characteristics for steady axially symmetric flow if v„ and 
y are replaced by v T and r, where r is the cylindrical co-ordinate (the distance from the axis 
of symmetry, which is the *-axis); it is clear that the derivation is unchanged if we consider, instead of 
the acy-plane, an xr-plane through the axis of symmetry. 



§109 



Characteristics in steady two-dimensional flow 



435 



The characteristics given by this equation do not depend on the particular 
solution of Chaplygin's equation considered, because the coefficients in that 
equation are independent of O. The characteristics in the hodograph plane 
are a transformation of the characteristics C + and C- in the physical plane, 
and we call them respectively the characteristics V + and T_, in accordance 
with the signs in (109.2). 

The integration of equation (109.2) gives relations of the form J + (v, d) 
= constant, J-(v, 6) = constant. The functions / + and /_ are quantities 
which remain constant along the characteristics C + and C- (i.e. Riemann 
invariants). For a perfect gas, equation (109.2) can be integrated explicitly. 




Fig. 100 



There is, however, no need to go through the calculations, since the result 
can be seen from formulae (107.3) and (107.4). For, according to the general 
properties of simple waves (see §97), the dependence of v on 6 for a simple 
wave is given